# Difference between revisions of "Talk:Main Page"

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− | === | + | === Sums of like powers=== |

− | + | ==== Sums of k m-th powers >= 0 (List A)==== | |

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{| class="wikitable" style="text-align:left" | {| class="wikitable" style="text-align:left" | ||

− | !   !! | + | !   !!m=2!!m=3!!m=4!!m=5!!m=6!!m=7!!m=8!!m=9!!m=10!! !! !!m=13 |

|- | |- | ||

− | | | + | | k>=2 ||<span title="Sums of at least 2 squares s'', for s >= 4.">[https://oeis.org/A176209 A176209]</span>|| || || || || || || || || || ||  |

|- | |- | ||

− | | | + | | k=-1 ||<span title="Sum of square displacements over all n-step self-avoiding walks on a 2D square lattice.">[https://oeis.org/A336448 A336448]</span>||<span title="Sum of the cubes of the parts in the partitions of n into two distinct parts.">[https://oeis.org/A294287 A294287]</span>||<span title="Sum of the fourth powers of the parts in the partitions of n into two distinct parts.">[https://oeis.org/A294288 A294288]</span>||<span title="Sum of the fifth powers of the parts in the partitions of n into two distinct parts.">[https://oeis.org/A294300 A294300]</span>||<span title="Sum of the sixth powers of the parts in the partitions of n into two distinct parts.">[https://oeis.org/A294301 A294301]</span>||<span title="Sum of the seventh powers of the parts in the partitions of n into two distinct parts.">[https://oeis.org/A294302 A294302]</span>||<span title="Numbers that are sums of 8th powers of 2 distinct positive integers.">[https://oeis.org/A155468 A155468]</span>||<span title="Sum of 9th powers.">[https://oeis.org/A007487 A007487]</span>||<span title="Sum of the tenth powers of the parts in the partitions of n into two distinct parts.">[https://oeis.org/A294305 A294305]</span>|| || ||<span title="Sum of 13th powers: 0^13+1^13+2^13+...+n^13.">[https://oeis.org/A181134 A181134]</span> |

|- | |- | ||

− | | | + | | k=2 ||<span title="Sum of two squares of Lucas numbers (A000032).">[https://oeis.org/A140328 A140328]</span>||<span title="Sums of two nonnegative cubes.">[https://oeis.org/A004999 A004999]</span>||<span title="Numbers that are the sum of two 4th powers in more than one way.">[https://oeis.org/A018786 A018786]</span>|| || || || || || || || ||  |

|- | |- | ||

− | | | + | | k=3 ||<span title="Numbers that are the sum of three squares (square 0 allowed) in exactly ten ways.">[https://oeis.org/A294713 A294713]</span>||<span title="Sum of three cubes problem: a(n) = integer x with the least possible absolute value such that n = x^3 + y^3 + z^3 with |x| >= |y| >= |z|, or 0 if no such x exists.">[https://oeis.org/A332201 A332201]</span>||<span title="Numbers that are the sum of three biquadrates (fourth powers) in more than one way.">[https://oeis.org/A193244 A193244]</span>|| || || || || || || || ||  |

|- | |- | ||

− | | | + | | k=4 || ||<span title="Numbers that are the sum of 4 cubes in more than 1 way.">[https://oeis.org/A001245 A001245]</span>|| || || || || || || || || ||  |

|- | |- | ||

− | | | + | |} |

+ | ==== Sums of exactly k positive m-th powers > 0 (List B)==== | ||

+ | {| class="wikitable" style="text-align:left" | ||

+ | !   !!m=2!!m=3!!m=4!!m=5!!m=6!!m=7!!m=8!!m=9!!m=10!!m=11 | ||

|- | |- | ||

− | | | + | | k=2 ||<span title="Numbers that are the sum of 2 nonzero squares, including repetitions.">[https://oeis.org/A024509 A024509]</span>||<span title="Numbers that are the sum of 2 positive cubes.">[https://oeis.org/A003325 A003325]</span>||<span title="Numbers that are the sum of 2 positive 4th powers.">[https://oeis.org/A003336 A003336]</span>||<span title="Numbers that are the sum of 2 positive 5th powers.">[https://oeis.org/A003347 A003347]</span>||<span title="Numbers that are the sum of 2 nonzero 6th powers.">[https://oeis.org/A003358 A003358]</span>||<span title="Numbers that are the sum of 2 positive 7th powers.">[https://oeis.org/A003369 A003369]</span>||<span title="Numbers that are the sum of 2 nonzero 8th powers.">[https://oeis.org/A003380 A003380]</span>||<span title="Numbers that are the sum of 2 positive 9th powers.">[https://oeis.org/A003391 A003391]</span>||<span title="Numbers that are the sum of 2 nonzero 10th powers.">[https://oeis.org/A004802 A004802]</span>||<span title="Numbers that are the sum of 2 positive 11th powers.">[https://oeis.org/A004813 A004813]</span> |

|- | |- | ||

− | | | + | | k=3 ||<span title="Numbers that are the sum of 3 nonzero squares, including repetitions.">[https://oeis.org/A024795 A024795]</span>||<span title="Numbers that are the sum of 3 positive cubes, including repetitions.">[https://oeis.org/A024981 A024981]</span>||<span title="Numbers that are the sum of 3 nonzero 4th powers in more than one way.">[https://oeis.org/A309762 A309762]</span>||<span title="Numbers that are the sum of 3 positive 5th powers.">[https://oeis.org/A003348 A003348]</span>||<span title="Numbers that are the sum of 3 nonzero 6th powers.">[https://oeis.org/A003359 A003359]</span>||<span title="Numbers that are the sum of 3 positive 7th powers.">[https://oeis.org/A003370 A003370]</span>||<span title="Numbers that are the sum of 3 nonzero 8th powers.">[https://oeis.org/A003381 A003381]</span>||<span title="Numbers that are the sum of 3 positive 9th powers.">[https://oeis.org/A003392 A003392]</span>||<span title="Numbers that are the sum of 3 nonzero 10th powers.">[https://oeis.org/A004803 A004803]</span>||<span title="Numbers that are the sum of 3 positive 11th powers.">[https://oeis.org/A004814 A004814]</span> |

|- | |- | ||

− | | | + | | k=4 ||<span title="Numbers that are the sum of 4 nonzero squares.">[https://oeis.org/A000414 A000414]</span>|| ||<span title="Numbers that are the sum of 4 nonzero 4th powers in more than one way.">[https://oeis.org/A309763 A309763]</span>||<span title="Numbers that are the sum of 4 positive 5th powers.">[https://oeis.org/A003349 A003349]</span>||<span title="Numbers that are the sum of 4 positive 6th powers.">[https://oeis.org/A003360 A003360]</span>||<span title="Numbers that are the sum of 4 positive 7th powers.">[https://oeis.org/A003371 A003371]</span>||<span title="Numbers that are the sum of 4 nonzero 8th powers.">[https://oeis.org/A003382 A003382]</span>||<span title="Numbers that are the sum of 4 positive 9th powers.">[https://oeis.org/A003393 A003393]</span>||<span title="Numbers that are the sum of 4 nonzero 10th powers.">[https://oeis.org/A004804 A004804]</span>||<span title="Numbers that are the sum of 4 positive 11th powers.">[https://oeis.org/A004815 A004815]</span> |

|- | |- | ||

− | | | + | | k=5 ||<span title="Numbers that are the sum of 5 positive squares.">[https://oeis.org/A047700 A047700]</span>||<span title="Numbers that are the sum of 5 positive cubes.">[https://oeis.org/A003328 A003328]</span>||<span title="Numbers that are the sum of 5 positive 4th powers.">[https://oeis.org/A003339 A003339]</span>||<span title="Numbers that are the sum of 5 positive 5th powers.">[https://oeis.org/A003350 A003350]</span>||<span title="Numbers that are the sum of 5 positive 6th powers.">[https://oeis.org/A003361 A003361]</span>||<span title="Numbers that are the sum of 5 positive 7th powers.">[https://oeis.org/A003372 A003372]</span>||<span title="Numbers that are the sum of 5 nonzero 8th powers.">[https://oeis.org/A003383 A003383]</span>||<span title="Numbers that are the sum of 5 positive 9th powers.">[https://oeis.org/A003394 A003394]</span>||<span title="Numbers that are the sum of 5 positive 10th powers.">[https://oeis.org/A004805 A004805]</span>||<span title="Numbers that are the sum of 5 positive 11th powers.">[https://oeis.org/A004816 A004816]</span> |

|- | |- | ||

− | | | + | | k=6 || ||<span title="Numbers that are the sum of 6 positive cubes.">[https://oeis.org/A003329 A003329]</span>||<span title="Numbers that are the sum of 6 positive 4th powers.">[https://oeis.org/A003340 A003340]</span>||<span title="Numbers that are the sum of 6 positive 5th powers.">[https://oeis.org/A003351 A003351]</span>||<span title="Numbers that are the sum of 6 positive 6th powers.">[https://oeis.org/A003362 A003362]</span>||<span title="Numbers that are the sum of 6 positive 7th powers.">[https://oeis.org/A003373 A003373]</span>||<span title="Numbers that are the sum of 6 nonzero 8th powers.">[https://oeis.org/A003384 A003384]</span>||<span title="Numbers that are the sum of 6 positive 9th powers.">[https://oeis.org/A003395 A003395]</span>||<span title="Numbers that are the sum of 6 positive 10th powers.">[https://oeis.org/A004806 A004806]</span>||<span title="Numbers that are the sum of 6 positive 11th powers.">[https://oeis.org/A004817 A004817]</span> |

+ | |- | ||

+ | | k=7 || ||<span title="Numbers that are the sum of 7 positive cubes.">[https://oeis.org/A003330 A003330]</span>||<span title="Numbers that are the sum of 7 positive 4th powers.">[https://oeis.org/A003341 A003341]</span>||<span title="Numbers that are the sum of 7 positive 5th powers.">[https://oeis.org/A003352 A003352]</span>||<span title="Numbers that are the sum of 7 positive 6th powers.">[https://oeis.org/A003363 A003363]</span>||<span title="Numbers that are the sum of 7 positive 7th powers.">[https://oeis.org/A003374 A003374]</span>||<span title="Numbers that are the sum of 7 nonzero 8th powers.">[https://oeis.org/A003385 A003385]</span>||<span title="Numbers that are the sum of 7 positive 9th powers.">[https://oeis.org/A003396 A003396]</span>||<span title="Numbers that are the sum of 7 positive 10th powers.">[https://oeis.org/A004807 A004807]</span>||<span title="Numbers that are the sum of 7 positive 11th powers.">[https://oeis.org/A004818 A004818]</span> | ||

+ | |- | ||

+ | | k=8 || ||<span title="Numbers that are the sum of 8 positive cubes.">[https://oeis.org/A003331 A003331]</span>||<span title="Numbers that are the sum of 8 positive 4th powers.">[https://oeis.org/A003342 A003342]</span>||<span title="Numbers that are the sum of 8 positive 5th powers.">[https://oeis.org/A003353 A003353]</span>||<span title="Numbers that are the sum of 8 positive 6th powers.">[https://oeis.org/A003364 A003364]</span>||<span title="Numbers that are the sum of 8 positive 7th powers.">[https://oeis.org/A003375 A003375]</span>||<span title="Numbers that are the sum of 8 nonzero 8th powers.">[https://oeis.org/A003386 A003386]</span>||<span title="Numbers that are the sum of 8 positive 9th powers.">[https://oeis.org/A003397 A003397]</span>||<span title="Numbers that are the sum of 8 positive 10th powers.">[https://oeis.org/A004808 A004808]</span>||<span title="Numbers that are the sum of 8 positive 11th powers.">[https://oeis.org/A004819 A004819]</span> | ||

+ | |- | ||

+ | | k=9 || ||<span title="Numbers that are the sum of 9 positive cubes.">[https://oeis.org/A003332 A003332]</span>||<span title="Numbers that are the sum of 9 positive 4th powers.">[https://oeis.org/A003343 A003343]</span>||<span title="Numbers that are the sum of 9 positive 5th powers.">[https://oeis.org/A003354 A003354]</span>||<span title="Numbers that are the sum of 9 positive 6th powers.">[https://oeis.org/A003365 A003365]</span>||<span title="Numbers that are the sum of 9 positive 7th powers.">[https://oeis.org/A003376 A003376]</span>||<span title="Numbers that are the sum of 9 nonzero 8th powers.">[https://oeis.org/A003387 A003387]</span>||<span title="Numbers that are the sum of 9 positive 9th powers.">[https://oeis.org/A003398 A003398]</span>||<span title="Numbers that are the sum of 9 positive 10th powers.">[https://oeis.org/A004809 A004809]</span>||<span title="Numbers that are the sum of 9 positive 11th powers.">[https://oeis.org/A004820 A004820]</span> | ||

+ | |- | ||

+ | | k=10 || ||<span title="Numbers that are the sum of 10 positive cubes.">[https://oeis.org/A003333 A003333]</span>||<span title="Numbers that are the sum of 10 positive 4th powers.">[https://oeis.org/A003344 A003344]</span>||<span title="Numbers that are the sum of 10 positive 5th powers.">[https://oeis.org/A003355 A003355]</span>||<span title="Numbers that are the sum of 10 positive 6th powers.">[https://oeis.org/A003366 A003366]</span>||<span title="Numbers that are the sum of 10 positive 7th powers.">[https://oeis.org/A003377 A003377]</span>||<span title="Sum of 10 nonzero 8th powers.">[https://oeis.org/A003388 A003388]</span>||<span title="Sum of 10 positive 9th powers.">[https://oeis.org/A003399 A003399]</span>||<span title="Numbers that are the sum of 10 positive 10th powers.">[https://oeis.org/A004810 A004810]</span>||<span title="Numbers that are the sum of 10 positive 11th powers.">[https://oeis.org/A004821 A004821]</span> | ||

+ | |- | ||

+ | | k=11 || ||<span title="Numbers that are the sum of 11 positive cubes.">[https://oeis.org/A003334 A003334]</span>||<span title="Numbers that are the sum of 11 positive 4th powers.">[https://oeis.org/A003345 A003345]</span>||<span title="Numbers that are the sum of 11 positive 5th powers.">[https://oeis.org/A003356 A003356]</span>||<span title="Numbers that are the sum of 11 positive 6th powers.">[https://oeis.org/A003367 A003367]</span>||<span title="Numbers that are the sum of 11 positive 7th powers.">[https://oeis.org/A003378 A003378]</span>||<span title="Numbers that are the sum of 11 positive 8th powers.">[https://oeis.org/A003389 A003389]</span>||<span title="Sum of 11 positive 9th powers.">[https://oeis.org/A004800 A004800]</span>||<span title="Numbers that are the sum of 11 positive 10th powers.">[https://oeis.org/A004811 A004811]</span>||<span title="Numbers that are the sum of 11 positive 11th powers.">[https://oeis.org/A004822 A004822]</span> | ||

+ | |- | ||

+ | | k=12 || ||<span title="Numbers that are the sum of 12 positive cubes.">[https://oeis.org/A003335 A003335]</span>||<span title="Numbers that are the sum of 12 positive 4th powers.">[https://oeis.org/A003346 A003346]</span>||<span title="Numbers that are the sum of 12 positive 5th powers.">[https://oeis.org/A003357 A003357]</span>||<span title="Numbers that are the sum of 12 positive 6th powers.">[https://oeis.org/A003368 A003368]</span>||<span title="Numbers that are the sum of 12 positive 7th powers.">[https://oeis.org/A003379 A003379]</span>||<span title="Sum of 12 nonzero 8th powers.">[https://oeis.org/A003390 A003390]</span>||<span title="Sum of 12 positive 9th powers.">[https://oeis.org/A004801 A004801]</span>||<span title="Numbers that are the sum of 12 positive 10th powers.">[https://oeis.org/A004812 A004812]</span>||<span title="Numbers that are the sum of 12 positive 11th powers.">[https://oeis.org/A004823 A004823]</span> | ||

+ | |- | ||

+ | | k=13 || || ||<span title="Sum of 13 nonzero fourth powers.">[https://oeis.org/A047724 A047724]</span>||<span title="Sum of 13 positive 5th powers.">[https://oeis.org/A123294 A123294]</span>|| || || || || ||  | ||

+ | |- | ||

+ | | k=14 || || ||<span title="Sum of 14 nonzero fourth powers.">[https://oeis.org/A047725 A047725]</span>||<span title="Sum of 14 positive 5th powers.">[https://oeis.org/A123295 A123295]</span>|| || || || || ||  | ||

+ | |- | ||

+ | |} | ||

+ | ==== Sums of at most k positive m-th powers > 0 (List C)==== | ||

+ | {| class="wikitable" style="text-align:left" | ||

+ | !   !!m=3!!m=4!!m=5!!m=6!!m=7!!m=8!!m=9!!m=10!!m=11 | ||

+ | |- | ||

+ | | k<=2 || ||<span title="Numbers that are the sum of at most 2 nonzero 4th powers.">[https://oeis.org/A004831 A004831]</span>||<span title="Numbers that are the sum of at most 2 positive 5th powers.">[https://oeis.org/A004842 A004842]</span>||<span title="Numbers that are the sum of at most 2 nonzero 6th powers.">[https://oeis.org/A004853 A004853]</span>||<span title="Numbers that are the sum of at most 2 positive 7th powers.">[https://oeis.org/A004864 A004864]</span>||<span title="Numbers that are the sum of at most 2 nonzero 8th powers.">[https://oeis.org/A004875 A004875]</span>||<span title="Numbers that are the sum of at most 2 positive 9th powers.">[https://oeis.org/A004886 A004886]</span>||<span title="Numbers that are the sum of at most 2 nonzero 10th powers.">[https://oeis.org/A004897 A004897]</span>||<span title="Numbers that are the sum of at most 2 positive 11th powers.">[https://oeis.org/A004908 A004908]</span> | ||

+ | |- | ||

+ | | k<=3 ||<span title="Numbers that are the sum of at most 3 positive cubes.">[https://oeis.org/A004825 A004825]</span>||<span title="Numbers that are the sum of at most 3 nonzero 4th powers.">[https://oeis.org/A004832 A004832]</span>||<span title="Numbers that are the sum of at most 3 positive 5th powers.">[https://oeis.org/A004843 A004843]</span>||<span title="Numbers that are the sum of at most 3 nonzero 6th powers.">[https://oeis.org/A004854 A004854]</span>||<span title="Numbers that are the sum of at most 3 positive 7th powers.">[https://oeis.org/A004865 A004865]</span>||<span title="Numbers that are the sum of at most 3 nonzero 8th powers.">[https://oeis.org/A004876 A004876]</span>||<span title="Numbers that are the sum of at most 3 positive 9th powers.">[https://oeis.org/A004887 A004887]</span>||<span title="Numbers that are the sum of at most 3 nonzero 10th powers.">[https://oeis.org/A004898 A004898]</span>||<span title="Numbers that are the sum of at most 3 positive 11th powers.">[https://oeis.org/A004909 A004909]</span> | ||

+ | |- | ||

+ | | k<=4 ||<span title="Numbers that are the sum of at most 4 positive cubes.">[https://oeis.org/A004826 A004826]</span>||<span title="Numbers that are the sum of at most 4 nonzero 4th powers.">[https://oeis.org/A004833 A004833]</span>||<span title="Numbers that are the sum of at most 4 positive 5th powers.">[https://oeis.org/A004844 A004844]</span>||<span title="Numbers that are the sum of at most 4 nonzero 6th powers.">[https://oeis.org/A004855 A004855]</span>||<span title="Numbers that are the sum of at most 4 positive 7th powers.">[https://oeis.org/A004866 A004866]</span>||<span title="Numbers that are the sum of at most 4 nonzero 8th powers.">[https://oeis.org/A004877 A004877]</span>||<span title="Numbers that are the sum of at most 4 positive 9th powers.">[https://oeis.org/A004888 A004888]</span>||<span title="Numbers that are the sum of at most 4 nonzero 10th powers.">[https://oeis.org/A004899 A004899]</span>||<span title="Numbers that are the sum of at most 4 positive 11th powers.">[https://oeis.org/A004910 A004910]</span> | ||

+ | |- | ||

+ | | k<=5 ||<span title="Numbers that are the sum of at most 5 positive cubes.">[https://oeis.org/A004827 A004827]</span>||<span title="Numbers that are the sum of at most 5 nonzero 4th powers.">[https://oeis.org/A004834 A004834]</span>||<span title="Numbers that are the sum of at most 5 positive 5th powers.">[https://oeis.org/A004845 A004845]</span>||<span title="Numbers that are the sum of at most 5 nonzero 6th powers.">[https://oeis.org/A004856 A004856]</span>||<span title="Numbers that are the sum of at most 5 positive 7th powers.">[https://oeis.org/A004867 A004867]</span>||<span title="Numbers that are the sum of at most 5 nonzero 8th powers.">[https://oeis.org/A004878 A004878]</span>||<span title="Numbers that are the sum of at most 5 positive 9th powers.">[https://oeis.org/A004889 A004889]</span>||<span title="Numbers that are the sum of at most 5 nonzero 10th powers.">[https://oeis.org/A004900 A004900]</span>||<span title="Numbers that are the sum of at most 5 positive 11th powers.">[https://oeis.org/A004911 A004911]</span> | ||

+ | |- | ||

+ | | k<=6 ||<span title="Numbers that are the sum of at most 6 positive cubes.">[https://oeis.org/A004828 A004828]</span>||<span title="Numbers that are the sum of at most 6 nonzero 4th powers.">[https://oeis.org/A004835 A004835]</span>||<span title="Numbers that are the sum of at most 6 positive 5th powers.">[https://oeis.org/A004846 A004846]</span>||<span title="Numbers that are the sum of at most 6 nonzero 6th powers.">[https://oeis.org/A004857 A004857]</span>||<span title="Numbers that are the sum of at most 6 positive 7th powers.">[https://oeis.org/A004868 A004868]</span>||<span title="Numbers that are the sum of at most 6 nonzero 8th powers.">[https://oeis.org/A004879 A004879]</span>||<span title="Numbers that are the sum of at most 6 positive 9th powers.">[https://oeis.org/A004890 A004890]</span>||<span title="Numbers that are the sum of at most 6 nonzero 10th powers.">[https://oeis.org/A004901 A004901]</span>||<span title="Numbers that are the sum of at most 6 positive 11th powers.">[https://oeis.org/A004912 A004912]</span> | ||

+ | |- | ||

+ | | k<=7 ||<span title="Numbers that are the sum of at most 7 positive cubes.">[https://oeis.org/A004829 A004829]</span>||<span title="Numbers that are the sum of at most 7 nonzero 4th powers.">[https://oeis.org/A004836 A004836]</span>||<span title="Numbers that are the sum of at most 7 positive 5th powers.">[https://oeis.org/A004847 A004847]</span>||<span title="Numbers that are the sum of at most 7 nonzero 6th powers.">[https://oeis.org/A004858 A004858]</span>||<span title="Numbers that are the sum of at most 7 positive 7th powers.">[https://oeis.org/A004869 A004869]</span>||<span title="Numbers that are the sum of at most 7 nonzero 8th powers.">[https://oeis.org/A004880 A004880]</span>||<span title="Numbers that are the sum of at most 7 positive 9th powers.">[https://oeis.org/A004891 A004891]</span>||<span title="Numbers that are the sum of at most 7 nonzero 10th powers.">[https://oeis.org/A004902 A004902]</span>||<span title="Numbers that are the sum of at most 7 positive 11th powers.">[https://oeis.org/A004913 A004913]</span> | ||

+ | |- | ||

+ | | k<=8 ||<span title="Numbers that are the sum of at most 8 positive cubes.">[https://oeis.org/A004830 A004830]</span>||<span title="Numbers that are the sum of at most 8 nonzero 4th powers.">[https://oeis.org/A004837 A004837]</span>||<span title="Numbers that are the sum of at most 8 positive 5th powers.">[https://oeis.org/A004848 A004848]</span>||<span title="Numbers that are the sum of at most 8 nonzero 6th powers.">[https://oeis.org/A004859 A004859]</span>||<span title="Numbers that are the sum of at most 8 positive 7th powers.">[https://oeis.org/A004870 A004870]</span>||<span title="Numbers that are the sum of at most 8 nonzero 8th powers.">[https://oeis.org/A004881 A004881]</span>||<span title="Numbers that are the sum of at most 8 positive 9th powers.">[https://oeis.org/A004892 A004892]</span>||<span title="Numbers that are the sum of at most 8 nonzero 10th powers.">[https://oeis.org/A004903 A004903]</span>||<span title="Numbers that are the sum of at most 8 positive 11th powers.">[https://oeis.org/A004914 A004914]</span> | ||

+ | |- | ||

+ | | k<=9 || ||<span title="Numbers that are the sum of at most 9 nonzero 4th powers.">[https://oeis.org/A004838 A004838]</span>||<span title="Numbers that are the sum of at most 9 positive 5th powers.">[https://oeis.org/A004849 A004849]</span>||<span title="Numbers that are the sum of at most 9 nonzero 6th powers.">[https://oeis.org/A004860 A004860]</span>||<span title="Numbers that are the sum of at most 9 positive 7th powers.">[https://oeis.org/A004871 A004871]</span>||<span title="Numbers that are the sum of at most 9 nonzero 8th powers.">[https://oeis.org/A004882 A004882]</span>||<span title="Numbers that are the sum of at most 9 positive 9th powers.">[https://oeis.org/A004893 A004893]</span>||<span title="Numbers that are the sum of at most 9 nonzero 10th powers.">[https://oeis.org/A004904 A004904]</span>||<span title="Numbers that are the sum of at most 9 positive 11th powers.">[https://oeis.org/A004915 A004915]</span> | ||

+ | |- | ||

+ | | k<=10 || ||<span title="Numbers that are the sum of at most 10 nonzero 4th powers.">[https://oeis.org/A004839 A004839]</span>||<span title="Numbers that are the sum of at most 10 positive 5th powers.">[https://oeis.org/A004850 A004850]</span>||<span title="Numbers that are the sum of at most 10 nonzero 6th powers.">[https://oeis.org/A004861 A004861]</span>||<span title="Numbers that are the sum of at most 10 positive 7th powers.">[https://oeis.org/A004872 A004872]</span>||<span title="Numbers that are the sum of at most 10 nonzero 8th powers.">[https://oeis.org/A004883 A004883]</span>||<span title="Numbers that are the sum of at most 10 positive 9th powers.">[https://oeis.org/A004894 A004894]</span>||<span title="Numbers that are the sum of at most 10 nonzero 10th powers.">[https://oeis.org/A004905 A004905]</span>||<span title="Numbers that are the sum of at most 10 positive 11th powers.">[https://oeis.org/A004916 A004916]</span> | ||

+ | |- | ||

+ | | k<=11 || ||<span title="Numbers that are the sum of at most 11 nonzero 4th powers.">[https://oeis.org/A004840 A004840]</span>||<span title="Numbers that are the sum of at most 11 positive 5th powers.">[https://oeis.org/A004851 A004851]</span>||<span title="Numbers that are the sum of at most 11 nonzero 6th powers.">[https://oeis.org/A004862 A004862]</span>||<span title="Numbers that are the sum of at most 11 positive 7th powers.">[https://oeis.org/A004873 A004873]</span>||<span title="Numbers that are the sum of at most 11 nonzero 8th powers.">[https://oeis.org/A004884 A004884]</span>||<span title="Numbers that are the sum of at most 11 positive 9th powers.">[https://oeis.org/A004895 A004895]</span>||<span title="Numbers that are the sum of at most 11 nonzero 10th powers.">[https://oeis.org/A004906 A004906]</span>||<span title="Numbers that are the sum of at most 11 positive 11th powers.">[https://oeis.org/A004917 A004917]</span> | ||

+ | |- | ||

+ | | k<=12 || ||<span title="Numbers that are the sum of at most 12 nonzero 4th powers.">[https://oeis.org/A004841 A004841]</span>||<span title="Numbers that are the sum of at most 12 positive 5th powers.">[https://oeis.org/A004852 A004852]</span>||<span title="Numbers that are the sum of at most 12 nonzero 6th powers.">[https://oeis.org/A004863 A004863]</span>||<span title="Numbers that are the sum of at most 12 positive 7th powers.">[https://oeis.org/A004874 A004874]</span>||<span title="Numbers that are the sum of at most 12 nonzero 8th powers.">[https://oeis.org/A004885 A004885]</span>||<span title="Numbers that are the sum of at most 12 positive 9th powers.">[https://oeis.org/A004896 A004896]</span>||<span title="Numbers that are the sum of at most 12 nonzero 10th powers.">[https://oeis.org/A004907 A004907]</span>||<span title="Numbers that are the sum of at most 12 positive 11th powers.">[https://oeis.org/A004918 A004918]</span> | ||

+ | |- | ||

+ | |} | ||

+ | ==== Sums of k positive m-th powers > 1 (List D)==== | ||

+ | {| class="wikitable" style="text-align:left" | ||

+ | !   !!m=2!!m=3 | ||

+ | |- | ||

+ | | k=-1 || ||<span title="Numbers which can be written as sum of cubes > 1.">[https://oeis.org/A078131 A078131]</span> | ||

+ | |- | ||

+ | | k=2 || ||<span title="Numbers that are the sum of 2 cubes > 1.">[https://oeis.org/A294073 A294073]</span> | ||

+ | |- | ||

+ | | k=3 ||<span title="Numbers that are the sum of 3 squares > 1.">[https://oeis.org/A302359 A302359]</span>||<span title="Numbers that are the sum of 3 cubes > 1.">[https://oeis.org/A302360 A302360]</span> | ||

+ | |- | ||

+ | |} | ||

+ | ==== Numbers that have exactly k representations as the sum of m squares >= 0 (List E)==== | ||

+ | <!--A295158 quant_eq ten representations as the sum of five least_0 pow_2. --> | ||

+ | {| class="wikitable" style="text-align:left" | ||

+ | !   !!m=2!! !! !!m=5!!m=6!!m=7 | ||

+ | |- | ||

+ | | k=1 || || || || ||<span title="Numbers that have exactly one representation as a sum of six nonnegative squares.">[https://oeis.org/A295484 A295484]</span>||  | ||

+ | |- | ||

+ | | k=2 ||<span title="Numbers that are the sum of 2 squares in exactly 2 ways.">[https://oeis.org/A085625 A085625]</span>|| || ||<span title="Numbers that have exactly two representations as a sum of five nonnegative squares.">[https://oeis.org/A295150 A295150]</span>||<span title="Numbers that have exactly two representations as a sum of six nonnegative squares.">[https://oeis.org/A295485 A295485]</span>||<span title="Numbers that have exactly two representations of a sum of seven nonnegative squares.">[https://oeis.org/A295742 A295742]</span> | ||

+ | |- | ||

+ | | k=3 ||<span title="Numbers that are the sum of 2 squares in exactly 3 ways.">[https://oeis.org/A000443 A000443]</span>|| || ||<span title="Numbers that have exactly three representations as a sum of five nonnegative squares.">[https://oeis.org/A295151 A295151]</span>||<span title="Numbers that have exactly three representations as a sum of six nonnegative squares.">[https://oeis.org/A295486 A295486]</span>||<span title="Numbers that have exactly three representations of a sum of seven nonnegative squares.">[https://oeis.org/A295743 A295743]</span> | ||

+ | |- | ||

+ | | k=4 || || || ||<span title="Numbers that have exactly four representations as a sum of five nonnegative squares.">[https://oeis.org/A295152 A295152]</span>||<span title="Numbers that have exactly four representations as a sum of six nonnegative squares.">[https://oeis.org/A295487 A295487]</span>||<span title="Numbers that have exactly four representations of a sum of seven nonnegative squares.">[https://oeis.org/A295744 A295744]</span> | ||

+ | |- | ||

+ | | k=5 ||<span title="Numbers that are the sum of 2 squares in exactly 5 ways.">[https://oeis.org/A294716 A294716]</span>|| || ||<span title="Numbers that have exactly five representations as a sum of five nonnegative squares.">[https://oeis.org/A295153 A295153]</span>||<span title="Numbers that have exactly five representations as a sum of six nonnegative squares.">[https://oeis.org/A295488 A295488]</span>||<span title="Numbers that have exactly five representations of a sum of seven nonnegative squares.">[https://oeis.org/A295745 A295745]</span> | ||

+ | |- | ||

+ | | k=6 || || || ||<span title="Numbers that have exactly six representations as a sum of five nonnegative squares.">[https://oeis.org/A295154 A295154]</span>||<span title="Numbers that have exactly six representations as a sum of six nonnegative squares.">[https://oeis.org/A295489 A295489]</span>||<span title="Numbers that have exactly six representations of a sum of seven nonnegative squares.">[https://oeis.org/A295747 A295747]</span> | ||

+ | |- | ||

+ | | k=7 || || || ||<span title="Numbers that have exactly seven representations as a sum of five nonnegative squares.">[https://oeis.org/A295155 A295155]</span>||<span title="Numbers that have exactly seven representations as a sum of six nonnegative squares.">[https://oeis.org/A295490 A295490]</span>||<span title="Numbers that have exactly seven representations of a sum of seven nonnegative squares.">[https://oeis.org/A295748 A295748]</span> | ||

+ | |- | ||

+ | | k=8 || || || ||<span title="Numbers that have exactly eight representations as a sum of five nonnegative squares.">[https://oeis.org/A295156 A295156]</span>||<span title="Numbers that have exactly eight representations as a sum of six nonnegative squares.">[https://oeis.org/A295491 A295491]</span>||<span title="Numbers that have exactly eight representations of a sum of seven nonnegative squares.">[https://oeis.org/A295749 A295749]</span> | ||

+ | |- | ||

+ | | k=9 || || || ||<span title="Numbers that have exactly nine representations as a sum of five nonnegative squares.">[https://oeis.org/A295157 A295157]</span>||<span title="Numbers that have exactly nine representations as a sum of six nonnegative squares.">[https://oeis.org/A295492 A295492]</span>||<span title="Numbers that have exactly nine representations of a sum of seven nonnegative squares.">[https://oeis.org/A295750 A295750]</span> | ||

+ | |- | ||

+ | | k=10 || || || ||<span title="Numbers that have exactly ten representations as a sum of five nonnegative squares.">[https://oeis.org/A295158 A295158]</span>||<span title="Numbers that have exactly ten representations as a sum of six nonnegative squares.">[https://oeis.org/A295493 A295493]</span>||<span title="Numbers that have exactly ten representations as a sum of seven nonnegative squares.">[https://oeis.org/A295751 A295751]</span> | ||

+ | |- | ||

+ | |} | ||

+ | ===New Lists=== | ||

+ | ==== Numbers that can be expressed as the sum of k squares in m or more ways (List R2)==== | ||

+ | {| class="wikitable" style="text-align:left" | ||

+ | !   !!m>=1!!m>=2!!m>=3!!m>=4!!m>=5!!m>=6!!m>=7!!m>=8!!m>=9!!m>=10 | ||

+ | |- | ||

+ | | k=1 ||<span title="The squares: a(n) = n^2.">[https://oeis.org/A000290 A000290]</span>|| || || || || || || || ||  | ||

+ | |- | ||

+ | | k=2 || ||<span title="Numbers that are the sum of 2 nonzero squares in 2 or more ways.">[https://oeis.org/A007692 A007692]</span>||<span title="Numbers that are the sum of 2 distinct nonzero squares in 3 or more ways.">[https://oeis.org/A025313 A025313]</span>||<span title="Numbers that are the sum of 2 distinct nonzero squares in 4 or more ways.">[https://oeis.org/A025314 A025314]</span>||<span title="Numbers that are the sum of 2 distinct nonzero squares in 5 or more ways.">[https://oeis.org/A025315 A025315]</span>||<span title="Numbers that are the sum of 2 distinct nonzero squares in 6 or more ways.">[https://oeis.org/A025316 A025316]</span>||<span title="Numbers that are the sum of 2 distinct nonzero squares in 7 or more ways.">[https://oeis.org/A025317 A025317]</span>||<span title="Numbers that are the sum of 2 distinct nonzero squares in 8 or more ways.">[https://oeis.org/A025318 A025318]</span>||<span title="Numbers that are the sum of 2 distinct nonzero squares in 9 or more ways.">[https://oeis.org/A025319 A025319]</span>||<span title="Numbers that are the sum of 2 distinct nonzero squares in 10 or more ways.">[https://oeis.org/A025320 A025320]</span> | ||

+ | |- | ||

+ | | k=3 ||<span title="Numbers expressible in more than one way as i^2 + j^2 + k^2, where 1 <= i <= j <= k.">[https://oeis.org/A024796 A024796]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in 2 or more ways.">[https://oeis.org/A024804 A024804]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in 3 or more ways.">[https://oeis.org/A025349 A025349]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in 4 or more ways.">[https://oeis.org/A025350 A025350]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in 5 or more ways.">[https://oeis.org/A025351 A025351]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in 6 or more ways.">[https://oeis.org/A025352 A025352]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in 7 or more ways.">[https://oeis.org/A025353 A025353]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in 8 or more ways.">[https://oeis.org/A025354 A025354]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in 9 or more ways.">[https://oeis.org/A025355 A025355]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in 10 or more ways.">[https://oeis.org/A025356 A025356]</span> | ||

+ | |- | ||

+ | | k=4 || ||<span title="Numbers that are representable in at least two ways as sums of four distinct nonvanishing squares.">[https://oeis.org/A259058 A259058]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in 3 or more ways.">[https://oeis.org/A025387 A025387]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in 4 or more ways.">[https://oeis.org/A025388 A025388]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in 5 or more ways.">[https://oeis.org/A025389 A025389]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in 6 or more ways.">[https://oeis.org/A025390 A025390]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in 7 or more ways.">[https://oeis.org/A025391 A025391]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in 8 or more ways.">[https://oeis.org/A025392 A025392]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in 9 or more ways.">[https://oeis.org/A025393 A025393]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in 10 or more ways.">[https://oeis.org/A025394 A025394]</span> | ||

+ | |- | ||

+ | | k=5 || ||<span title="Numbers that are the sum of five squares in two or more ways.">[https://oeis.org/A344795 A344795]</span>||<span title="Numbers that are the sum of five squares in three or more ways.">[https://oeis.org/A344796 A344796]</span>||<span title="Numbers that are the sum of five squares in four or more ways.">[https://oeis.org/A344797 A344797]</span>||<span title="Numbers that are the sum of five squares in five or more ways.">[https://oeis.org/A344798 A344798]</span>||<span title="Numbers that are the sum of five squares in six or more ways.">[https://oeis.org/A344799 A344799]</span>||<span title="Numbers that are the sum of five squares in seven or more ways.">[https://oeis.org/A344800 A344800]</span>||<span title="Numbers that are the sum of five squares in eight or more ways.">[https://oeis.org/A344801 A344801]</span>||<span title="Numbers that are the sum of five squares in nine or more ways.">[https://oeis.org/A344802 A344802]</span>||<span title="Numbers that are the sum of five squares in ten or more ways.">[https://oeis.org/A344803 A344803]</span> | ||

+ | |- | ||

+ | | k=6 ||<span title="Numbers that are the sum of six second powers in one or more ways.">[https://oeis.org/A344805 A344805]</span>||<span title="Numbers that are the sum of six squares in two or more ways.">[https://oeis.org/A344806 A344806]</span>||<span title="Numbers that are the sum of six squares in three or more ways.">[https://oeis.org/A344807 A344807]</span>||<span title="Numbers that are the sum of six squares in four or more ways.">[https://oeis.org/A344808 A344808]</span>||<span title="Numbers that are the sum of six squares in five or more ways.">[https://oeis.org/A344809 A344809]</span>||<span title="Numbers that are the sum of six squares in six or more ways.">[https://oeis.org/A344810 A344810]</span>||<span title="Numbers that are the sum of six squares in seven or more ways.">[https://oeis.org/A344811 A344811]</span>||<span title="Numbers that are the sum of six squares in eight or more ways.">[https://oeis.org/A344812 A344812]</span>||<span title="Numbers that are the sum of six squares in nine or more ways.">[https://oeis.org/A345476 A345476]</span>||<span title="Numbers that are the sum of six squares in ten or more ways.">[https://oeis.org/A345477 A345477]</span> | ||

+ | |- | ||

+ | | k=7 ||<span title="Numbers that are the sum of seven squares in one or more ways.">[https://oeis.org/A345478 A345478]</span>||<span title="Numbers that are the sum of seven squares in two or more ways.">[https://oeis.org/A345479 A345479]</span>||<span title="Numbers that are the sum of seven squares in three or more ways.">[https://oeis.org/A345480 A345480]</span>||<span title="Numbers that are the sum of seven squares in four or more ways.">[https://oeis.org/A345481 A345481]</span>||<span title="Numbers that are the sum of seven squares in five or more ways.">[https://oeis.org/A345482 A345482]</span>||<span title="Numbers that are the sum of seven squares in six or more ways.">[https://oeis.org/A345483 A345483]</span>||<span title="Numbers that are the sum of seven squares in seven or more ways.">[https://oeis.org/A345484 A345484]</span>||<span title="Numbers that are the sum of seven squares in eight or more ways.">[https://oeis.org/A345485 A345485]</span>||<span title="Numbers that are the sum of seven squares in nine or more ways.">[https://oeis.org/A345486 A345486]</span>||<span title="Numbers that are the sum of seven squares in ten or more ways.">[https://oeis.org/A345487 A345487]</span> | ||

+ | |- | ||

+ | | k=8 ||<span title="Numbers that are the sum of eight squares in one or more ways.">[https://oeis.org/A345488 A345488]</span>||<span title="Numbers that are the sum of eight squares in two or more ways.">[https://oeis.org/A345489 A345489]</span>||<span title="Numbers that are the sum of eight squares in three or more ways.">[https://oeis.org/A345490 A345490]</span>||<span title="Numbers that are the sum of eight squares in four or more ways.">[https://oeis.org/A345491 A345491]</span>||<span title="Numbers that are the sum of eight squares in five or more ways.">[https://oeis.org/A345492 A345492]</span>||<span title="Numbers that are the sum of eight squares in six or more ways.">[https://oeis.org/A345493 A345493]</span>||<span title="Numbers that are the sum of eight squares in seven or more ways.">[https://oeis.org/A345494 A345494]</span>||<span title="Numbers that are the sum of eight squares in eight or more ways.">[https://oeis.org/A345495 A345495]</span>||<span title="Numbers that are the sum of eight squares in nine or more ways.">[https://oeis.org/A345496 A345496]</span>||<span title="Numbers that are the sum of eight squares in ten or more ways.">[https://oeis.org/A345497 A345497]</span> | ||

+ | |- | ||

+ | | k=9 ||<span title="Numbers that are the sum of nine squares in one or more ways.">[https://oeis.org/A345498 A345498]</span>||<span title="Numbers that are the sum of nine squares in two or more ways.">[https://oeis.org/A345499 A345499]</span>||<span title="Numbers that are the sum of nine squares in three or more ways.">[https://oeis.org/A345500 A345500]</span>||<span title="Numbers that are the sum of nine squares in four or more ways.">[https://oeis.org/A345501 A345501]</span>||<span title="Numbers that are the sum of nine squares in five or more ways.">[https://oeis.org/A345502 A345502]</span>||<span title="Numbers that are the sum of nine squares in six or more ways.">[https://oeis.org/A345503 A345503]</span>||<span title="Numbers that are the sum of nine squares in seven or more ways.">[https://oeis.org/A345504 A345504]</span>||<span title="Numbers that are the sum of nine squares in eight or more ways.">[https://oeis.org/A345505 A345505]</span>|| ||<span title="Numbers that are the sum of nine squares in ten or more ways.">[https://oeis.org/A346803 A346803]</span> | ||

+ | |- | ||

+ | | k=10 ||<span title="Numbers that are the sum of ten squares in one or more ways.">[https://oeis.org/A345508 A345508]</span>||<span title="Numbers that are the sum of ten squares in two or more ways.">[https://oeis.org/A345509 A345509]</span>||<span title="Numbers that are the sum of ten squares in three or more ways.">[https://oeis.org/A345510 A345510]</span>|| ||<span title="Numbers that are the sum of ten squares in five or more ways.">[https://oeis.org/A346804 A346804]</span>||<span title="Numbers that are the sum of ten squares in six or more ways.">[https://oeis.org/A346805 A346805]</span>||<span title="Numbers that are the sum of ten squares in seven or more ways.">[https://oeis.org/A346806 A346806]</span>||<span title="Numbers that are the sum of ten squares in eight or more ways.">[https://oeis.org/A346807 A346807]</span>|| ||<span title="Numbers that are the sum of ten squares in ten or more ways.">[https://oeis.org/A346808 A346808]</span> | ||

+ | |- | ||

+ | |} | ||

+ | ==== Numbers that can be expressed as the sum of k squares in exactly m ways (List S2)==== | ||

+ | {| class="wikitable" style="text-align:left" | ||

+ | !   !!m=1!!m=2!!m=3!!m=4!!m=5!!m=6!!m=7!!m=8!!m=9!!m=10!!m=11 | ||

+ | |- | ||

+ | | k=2 ||<span title="Numbers that are the sum of 2 distinct nonzero squares in exactly 1 way.">[https://oeis.org/A025302 A025302]</span>||<span title="Numbers that are the sum of 2 squares in exactly 2 ways.">[https://oeis.org/A085625 A085625]</span>||<span title="Numbers that are the sum of 2 distinct nonzero squares in exactly 3 ways.">[https://oeis.org/A025304 A025304]</span>||<span title="Numbers that are the sum of 2 distinct nonzero squares in exactly 4 ways.">[https://oeis.org/A025305 A025305]</span>||<span title="Numbers that are the sum of 2 squares in exactly 5 ways.">[https://oeis.org/A294716 A294716]</span>||<span title="Numbers that are the sum of 2 distinct nonzero squares in exactly 6 ways.">[https://oeis.org/A025307 A025307]</span>||<span title="Numbers that are the sum of 2 distinct nonzero squares in exactly 7 ways.">[https://oeis.org/A025308 A025308]</span>||<span title="Numbers that are the sum of 2 distinct nonzero squares in exactly 8 ways.">[https://oeis.org/A025309 A025309]</span>||<span title="Numbers that are the sum of 2 distinct nonzero squares in exactly 9 ways.">[https://oeis.org/A025310 A025310]</span>||<span title="Numbers that are the sum of 2 distinct nonzero squares in exactly 10 ways.">[https://oeis.org/A025311 A025311]</span>||<span title="Numbers that are the sum of 2 nonzero squares in exactly 11 ways.">[https://oeis.org/A236711 A236711]</span> | ||

+ | |- | ||

+ | | k=3 ||<span title="Numbers that are the sum of 3 distinct nonzero squares in exactly one way.">[https://oeis.org/A025339 A025339]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in exactly 2 ways.">[https://oeis.org/A025340 A025340]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in exactly 3 ways.">[https://oeis.org/A025341 A025341]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in exactly 4 ways.">[https://oeis.org/A025342 A025342]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in exactly 5 ways.">[https://oeis.org/A025343 A025343]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in exactly 6 ways.">[https://oeis.org/A025344 A025344]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in exactly 7 ways.">[https://oeis.org/A025345 A025345]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in exactly 8 ways.">[https://oeis.org/A025346 A025346]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in exactly 9 ways.">[https://oeis.org/A025347 A025347]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in exactly 10 ways.">[https://oeis.org/A025348 A025348]</span>||  | ||

+ | |- | ||

+ | | k=4 ||<span title="Numbers that are the sum of 4 distinct nonzero squares in exactly 1 way.">[https://oeis.org/A025376 A025376]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in exactly 2 ways.">[https://oeis.org/A025377 A025377]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in exactly 3 ways.">[https://oeis.org/A025378 A025378]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in exactly 4 ways.">[https://oeis.org/A025379 A025379]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in exactly 5 ways.">[https://oeis.org/A025380 A025380]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in exactly 6 ways.">[https://oeis.org/A025381 A025381]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in exactly 7 ways.">[https://oeis.org/A025382 A025382]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in exactly 8 ways.">[https://oeis.org/A025383 A025383]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in exactly 9 ways.">[https://oeis.org/A025384 A025384]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in exactly 10 ways.">[https://oeis.org/A025385 A025385]</span>||  | ||

+ | |- | ||

+ | | k=5 ||<span title="Numbers that are the sum of 5 nonzero squares in exactly 1 way.">[https://oeis.org/A294675 A294675]</span>||<span title="Numbers that have exactly two representations as a sum of five nonnegative squares.">[https://oeis.org/A295150 A295150]</span>||<span title="Numbers that have exactly three representations as a sum of five nonnegative squares.">[https://oeis.org/A295151 A295151]</span>||<span title="Numbers that have exactly four representations as a sum of five nonnegative squares.">[https://oeis.org/A295152 A295152]</span>||<span title="Numbers that have exactly five representations as a sum of five nonnegative squares.">[https://oeis.org/A295153 A295153]</span>||<span title="Numbers that have exactly six representations as a sum of five nonnegative squares.">[https://oeis.org/A295154 A295154]</span>||<span title="Numbers that have exactly seven representations as a sum of five nonnegative squares.">[https://oeis.org/A295155 A295155]</span>||<span title="Numbers that have exactly eight representations as a sum of five nonnegative squares.">[https://oeis.org/A295156 A295156]</span>||<span title="Numbers that have exactly nine representations as a sum of five nonnegative squares.">[https://oeis.org/A295157 A295157]</span>||<span title="Numbers that have exactly ten representations as a sum of five nonnegative squares.">[https://oeis.org/A295158 A295158]</span>||  | ||

+ | |- | ||

+ | | k=6 ||<span title="Numbers that have exactly one representation as a sum of six positive squares.">[https://oeis.org/A295670 A295670]</span>||<span title="Numbers that have exactly two representations as a sum of six positive squares.">[https://oeis.org/A295692 A295692]</span>||<span title="Numbers that have exactly three representations as a sum of six positive squares.">[https://oeis.org/A295693 A295693]</span>||<span title="Numbers that have exactly four representations as a sum of six positive squares.">[https://oeis.org/A295694 A295694]</span>||<span title="Numbers that have exactly five representations as a sum of six positive squares.">[https://oeis.org/A295695 A295695]</span>||<span title="Numbers that have exactly six representations as a sum of six positive squares.">[https://oeis.org/A295696 A295696]</span>||<span title="Numbers that have exactly seven representations as a sum of six positive squares.">[https://oeis.org/A295697 A295697]</span>||<span title="Numbers that have exactly eight representations as a sum of six positive squares.">[https://oeis.org/A295698 A295698]</span>||<span title="Numbers that have exactly nine representations as a sum of six positive squares.">[https://oeis.org/A295699 A295699]</span>||<span title="Numbers that have exactly ten representations as a sum of six positive squares.">[https://oeis.org/A295700 A295700]</span>||  | ||

+ | |- | ||

+ | | k=7 ||<span title="Numbers that have exactly one representation as a sum of seven positive squares.">[https://oeis.org/A295797 A295797]</span>||<span title="Numbers that have exactly two representations as a sum of seven positive squares.">[https://oeis.org/A295799 A295799]</span>||<span title="Numbers that have exactly three representations as a sum of seven positive squares.">[https://oeis.org/A295800 A295800]</span>||<span title="Numbers that have exactly four representations as a sum of seven positive squares.">[https://oeis.org/A295801 A295801]</span>||<span title="Numbers that have exactly five representations as a sum of seven positive squares.">[https://oeis.org/A295802 A295802]</span>||<span title="Numbers that have exactly six representations as a sum of seven positive squares.">[https://oeis.org/A295803 A295803]</span>||<span title="Numbers that have exactly seven representations as a sum of seven positive squares.">[https://oeis.org/A295804 A295804]</span>||<span title="Numbers that have exactly eight representations as a sum of seven positive squares.">[https://oeis.org/A295805 A295805]</span>||<span title="Numbers that have exactly nine representations as a sum of seven positive squares.">[https://oeis.org/A295806 A295806]</span>||<span title="Numbers that have exactly ten representations as a sum of seven positive squares.">[https://oeis.org/A295807 A295807]</span>||  | ||

+ | |- | ||

+ | |} | ||

+ | ==== Numbers that can be expressed as the sum of k cubes in m or more ways (List R3)==== | ||

+ | {| class="wikitable" style="text-align:left" | ||

+ | !   !!m>=1!!m>=2!!m>=3!!m>=4!!m>=5!!m>=6!!m>=7!!m>=8!!m>=9!!m>=10 | ||

+ | |- | ||

+ | | k=1 ||<span title="The cubes: a(n) = n^3.">[https://oeis.org/A000578 A000578]</span>|| || || || || || || || ||  | ||

+ | |- | ||

+ | | k=2 ||<span title="Taxi-cab numbers: sums of 2 cubes in more than 1 way.">[https://oeis.org/A001235 A001235]</span>|| ||<span title="Numbers that are the sum of two positive cubes in at least three ways (all solutions).">[https://oeis.org/A018787 A018787]</span>||<span title="Numbers that are the sum of two positive cubes in at least four ways (all solutions).">[https://oeis.org/A023051 A023051]</span>||<span title="Sum of two positive cubes in at least five ways (all solutions).">[https://oeis.org/A051167 A051167]</span>|| || || || ||  | ||

+ | |- | ||

+ | | k=3 || ||<span title="Numbers that are the sum of 3 distinct positive cubes in 2 or more ways.">[https://oeis.org/A024974 A024974]</span>||<span title="Numbers that are the sum of 3 distinct positive cubes in 3 or more ways.">[https://oeis.org/A025402 A025402]</span>||<span title="Numbers that are the sum of three positive cubes in four or more ways.">[https://oeis.org/A343968 A343968]</span>||<span title="Numbers that are the sum of three positive cubes in five or more ways.">[https://oeis.org/A343967 A343967]</span>||<span title="Numbers that are the sum of three third powers in six or more ways.">[https://oeis.org/A345083 A345083]</span>||<span title="Numbers that are the sum of three third powers in seven or more ways.">[https://oeis.org/A345086 A345086]</span>||<span title="Numbers that are the sum of three third powers in eight or more ways.">[https://oeis.org/A345087 A345087]</span>||<span title="Numbers that are the sum of three third powers in nine or more ways.">[https://oeis.org/A345119 A345119]</span>||<span title="Numbers that are the sum of three third powers in ten or more ways.">[https://oeis.org/A345121 A345121]</span> | ||

+ | |- | ||

+ | | k=4 ||<span title="Numbers that are the sum of 4 positive cubes in 1 or more way.">[https://oeis.org/A003327 A003327]</span>||<span title="Numbers that are representable in at least two ways as sums of four distinct nonvanishing cubes.">[https://oeis.org/A259060 A259060]</span>||<span title="Numbers that are the sum of 4 distinct positive cubes in 3 or more ways.">[https://oeis.org/A025413 A025413]</span>||<span title="Numbers that are the sum of four positive cubes in four or more ways.">[https://oeis.org/A343971 A343971]</span>||<span title="Numbers that are the sum of four positive cubes in five or more ways.">[https://oeis.org/A343987 A343987]</span>||<span title="Numbers that are the sum of four third powers in six or more ways.">[https://oeis.org/A345148 A345148]</span>||<span title="Numbers that are the sum of four third powers in seven or more ways.">[https://oeis.org/A345150 A345150]</span>||<span title="Numbers that are the sum of four third powers in eight or more ways.">[https://oeis.org/A345152 A345152]</span>||<span title="Numbers that are the sum of four third powers in nine or more ways.">[https://oeis.org/A345146 A345146]</span>||<span title="Numbers that are the sum of four third powers in ten or more ways.">[https://oeis.org/A345155 A345155]</span> | ||

+ | |- | ||

+ | | k=5 || ||<span title="Numbers that are the sum of five positive cubes in two or more ways.">[https://oeis.org/A343702 A343702]</span>||<span title="Numbers that are the sum of five positive cubes in three or more ways.">[https://oeis.org/A343704 A343704]</span>||<span title="Numbers that are the sum of five positive cubes in four or more ways.">[https://oeis.org/A344034 A344034]</span>||<span title="Numbers that are the sum of five positive cubes in five or more ways.">[https://oeis.org/A343989 A343989]</span>||<span title="Numbers that are the sum of five third powers in six or more ways.">[https://oeis.org/A345174 A345174]</span>||<span title="Numbers that are the sum of five third powers in seven or more ways.">[https://oeis.org/A345180 A345180]</span>||<span title="Numbers that are the sum of five third powers in eight or more ways.">[https://oeis.org/A345183 A345183]</span>||<span title="Numbers that are the sum of five third powers in nine or more ways.">[https://oeis.org/A345185 A345185]</span>||<span title="Numbers that are the sum of five third powers in ten or more ways.">[https://oeis.org/A345187 A345187]</span> | ||

+ | |- | ||

+ | | k=6 || ||<span title="Numbers that are the sum of six cubes in two or more ways.">[https://oeis.org/A345511 A345511]</span>||<span title="Numbers that are the sum of six cubes in three or more ways.">[https://oeis.org/A345512 A345512]</span>||<span title="Numbers that are the sum of six cubes in four or more ways.">[https://oeis.org/A345513 A345513]</span>||<span title="Numbers that are the sum of six cubes in five or more ways.">[https://oeis.org/A345514 A345514]</span>||<span title="Numbers that are the sum of six cubes in six or more ways.">[https://oeis.org/A345515 A345515]</span>||<span title="Numbers that are the sum of six cubes in seven or more ways.">[https://oeis.org/A345516 A345516]</span>||<span title="Numbers that are the sum of six cubes in eight or more ways.">[https://oeis.org/A345517 A345517]</span>||<span title="Numbers that are the sum of six cubes in nine or more ways.">[https://oeis.org/A345518 A345518]</span>||<span title="Numbers that are the sum of six cubes in ten or more ways.">[https://oeis.org/A345519 A345519]</span> | ||

+ | |- | ||

+ | | k=7 || ||<span title="Numbers that are the sum of seven cubes in two or more ways.">[https://oeis.org/A345520 A345520]</span>||<span title="Numbers that are the sum of seven cubes in three or more ways.">[https://oeis.org/A345521 A345521]</span>||<span title="Numbers that are the sum of seven cubes in four or more ways.">[https://oeis.org/A345522 A345522]</span>||<span title="Numbers that are the sum of seven cubes in five or more ways.">[https://oeis.org/A345523 A345523]</span>||<span title="Numbers that are the sum of seven cubes in six or more ways.">[https://oeis.org/A345524 A345524]</span>||<span title="Numbers that are the sum of seven cubes in seven or more ways.">[https://oeis.org/A345525 A345525]</span>||<span title="Numbers that are the sum of seven cubes in eight or more ways.">[https://oeis.org/A345526 A345526]</span>||<span title="Numbers that are the sum of seven cubes in nine or more ways.">[https://oeis.org/A345527 A345527]</span>||<span title="Numbers that are the sum of seven cubes in ten or more ways.">[https://oeis.org/A345506 A345506]</span> | ||

+ | |- | ||

+ | | k=8 || ||<span title="Numbers that are the sum of eight cubes in two or more ways.">[https://oeis.org/A345532 A345532]</span>||<span title="Numbers that are the sum of eight cubes in three or more ways.">[https://oeis.org/A345533 A345533]</span>||<span title="Numbers that are the sum of eight cubes in four or more ways.">[https://oeis.org/A345534 A345534]</span>||<span title="Numbers that are the sum of eight cubes in five or more ways.">[https://oeis.org/A345535 A345535]</span>||<span title="Numbers that are the sum of eight cubes in six or more ways.">[https://oeis.org/A345536 A345536]</span>||<span title="Numbers that are the sum of eight cubes in seven or more ways.">[https://oeis.org/A345537 A345537]</span>||<span title="Numbers that are the sum of eight cubes in eight or more ways.">[https://oeis.org/A345538 A345538]</span>||<span title="Numbers that are the sum of eight cubes in nine or more ways.">[https://oeis.org/A345539 A345539]</span>||<span title="Numbers that are the sum of eight cubes in ten or more ways.">[https://oeis.org/A345540 A345540]</span> | ||

+ | |- | ||

+ | | k=9 || ||<span title="Numbers that are the sum of nine cubes in two or more ways.">[https://oeis.org/A345541 A345541]</span>||<span title="Numbers that are the sum of nine cubes in three or more ways.">[https://oeis.org/A345542 A345542]</span>||<span title="Numbers that are the sum of nine cubes in four or more ways.">[https://oeis.org/A345543 A345543]</span>||<span title="Numbers that are the sum of nine cubes in five or more ways.">[https://oeis.org/A345544 A345544]</span>||<span title="Numbers that are the sum of nine cubes in six or more ways.">[https://oeis.org/A345545 A345545]</span>||<span title="Numbers that are the sum of nine cubes in seven or more ways.">[https://oeis.org/A345546 A345546]</span>||<span title="Numbers that are the sum of nine cubes in eight or more ways.">[https://oeis.org/A345547 A345547]</span>||<span title="Numbers that are the sum of nine cubes in nine or more ways.">[https://oeis.org/A345548 A345548]</span>||<span title="Numbers that are the sum of nine cubes in ten or more ways.">[https://oeis.org/A345549 A345549]</span> | ||

+ | |- | ||

+ | | k=10 || ||<span title="Numbers that are the sum of ten cubes in two or more ways.">[https://oeis.org/A345550 A345550]</span>||<span title="Numbers that are the sum of ten cubes in three or more ways.">[https://oeis.org/A345551 A345551]</span>||<span title="Numbers that are the sum of ten cubes in four or more ways.">[https://oeis.org/A345552 A345552]</span>||<span title="Numbers that are the sum of ten cubes in five or more ways.">[https://oeis.org/A345553 A345553]</span>||<span title="Numbers that are the sum of ten cubes in six or more ways.">[https://oeis.org/A345554 A345554]</span>||<span title="Numbers that are the sum of ten cubes in seven or more ways.">[https://oeis.org/A345555 A345555]</span>||<span title="Numbers that are the sum of ten cubes in eight or more ways.">[https://oeis.org/A345556 A345556]</span>||<span title="Numbers that are the sum of ten cubes in nine or more ways.">[https://oeis.org/A345557 A345557]</span>||<span title="Numbers that are the sum of ten cubes in ten or more ways.">[https://oeis.org/A345558 A345558]</span> | ||

+ | |- | ||

+ | |} | ||

+ | ==== Numbers that can be expressed as the sum of k cubes in exactly m ways (List S3)==== | ||

+ | {| class="wikitable" style="text-align:left" | ||

+ | !   !!m=1!!m=2!!m=3!!m=4!!m=5!!m=6!!m=7!!m=8!!m=9!!m=10 | ||

+ | |- | ||

+ | | k=2 ||<span title="Numbers that are the sum of two positive cubes in exactly one way.">[https://oeis.org/A338667 A338667]</span>||<span title="Numbers that are the sum of two positive cubes in exactly two ways.">[https://oeis.org/A343708 A343708]</span>||<span title="Numbers that are the sum of two cubes in exactly three ways.">[https://oeis.org/A344804 A344804]</span>||<span title="Numbers that are the sum of two cubes in exactly four ways.">[https://oeis.org/A345865 A345865]</span>|| || || || || ||  | ||

+ | |- | ||

+ | | k=3 ||<span title="Numbers that are the sum of 3 distinct positive cubes in exactly 1 way.">[https://oeis.org/A025399 A025399]</span>||<span title="Numbers that are the sum of 3 distinct positive cubes in exactly 2 ways.">[https://oeis.org/A025400 A025400]</span>||<span title="Numbers that are the sum of 3 distinct positive cubes in exactly 3 ways.">[https://oeis.org/A025401 A025401]</span>||<span title="Numbers that are the sum of three positive cubes in exactly 4 ways.">[https://oeis.org/A343969 A343969]</span>||<span title="Numbers that are the sum of three positive cubes in exactly five ways.">[https://oeis.org/A343970 A343970]</span>||<span title="Numbers that are the sum of three third powers in exactly six ways.">[https://oeis.org/A345084 A345084]</span>||<span title="Numbers that are the sum of three third powers in exactly seven ways.">[https://oeis.org/A345085 A345085]</span>||<span title="Numbers that are the sum of three third powers in exactly eight ways.">[https://oeis.org/A345088 A345088]</span>||<span title="Numbers that are the sum of three third powers in exactly nine ways.">[https://oeis.org/A345120 A345120]</span>||<span title="Numbers that are the sum of three third powers in exactly ten ways.">[https://oeis.org/A345122 A345122]</span> | ||

+ | |- | ||

+ | | k=4 ||<span title="Numbers that are the sum of 4 distinct positive cubes in exactly 1 way.">[https://oeis.org/A025408 A025408]</span>||<span title="Numbers that are the sum of 4 distinct positive cubes in exactly 2 ways.">[https://oeis.org/A025409 A025409]</span>||<span title="Numbers that are the sum of 4 distinct positive cubes in exactly 3 ways.">[https://oeis.org/A025410 A025410]</span>||<span title="Numbers that are the sum of four positive cubes in exactly four ways.">[https://oeis.org/A343972 A343972]</span>||<span title="Numbers that are the sum of four positive cubes in exactly five ways.">[https://oeis.org/A343988 A343988]</span>||<span title="Numbers that are the sum of four third powers in exactly six ways.">[https://oeis.org/A345149 A345149]</span>||<span title="Numbers that are the sum of four third powers in exactly seven ways.">[https://oeis.org/A345151 A345151]</span>||<span title="Numbers that are the sum of four third powers in exactly eight ways.">[https://oeis.org/A345153 A345153]</span>||<span title="Numbers that are the sum of four third powers in exactly nine ways.">[https://oeis.org/A345154 A345154]</span>||<span title="Numbers that are the sum of four third powers in exactly ten ways.">[https://oeis.org/A345156 A345156]</span> | ||

+ | |- | ||

+ | | k=5 ||<span title="Numbers that are the sum of 5 positive cubes in exactly 1 way.">[https://oeis.org/A048926 A048926]</span>||<span title="Numbers that are the sum of 5 positive cubes in exactly 2 ways.">[https://oeis.org/A048927 A048927]</span>||<span title="Numbers that are the sum of five positive cubes in exactly three ways.">[https://oeis.org/A343705 A343705]</span>||<span title="Numbers that are the sum of five positive cubes in exactly four ways.">[https://oeis.org/A344035 A344035]</span>|| ||<span title="Numbers that are the sum of five third powers in exactly six ways.">[https://oeis.org/A345175 A345175]</span>||<span title="Numbers that are the sum of five third powers in exactly seven ways.">[https://oeis.org/A345181 A345181]</span>||<span title="Numbers that are the sum of five third powers in exactly eight ways.">[https://oeis.org/A345184 A345184]</span>||<span title="Numbers that are the sum of five third powers in exactly nine ways.">[https://oeis.org/A345186 A345186]</span>||<span title="Numbers that are the sum of five third powers in exactly ten ways.">[https://oeis.org/A345188 A345188]</span> | ||

+ | |- | ||

+ | | k=6 ||<span title="Numbers that are the sum of 6 positive cubes in exactly 1 way.">[https://oeis.org/A048929 A048929]</span>||<span title="Numbers that are the sum of 6 positive cubes in exactly 2 ways.">[https://oeis.org/A048930 A048930]</span>||<span title="Numbers that are the sum of 6 positive cubes in exactly 3 ways.">[https://oeis.org/A048931 A048931]</span>||<span title="Numbers that are the sum of six cubes in exactly four ways.">[https://oeis.org/A345766 A345766]</span>||<span title="Numbers that are the sum of six cubes in exactly five ways.">[https://oeis.org/A345767 A345767]</span>||<span title="Numbers that are the sum of six cubes in exactly six ways.">[https://oeis.org/A345768 A345768]</span>||<span title="Numbers that are the sum of six cubes in exactly seven ways.">[https://oeis.org/A345769 A345769]</span>||<span title="Numbers that are the sum of six cubes in exactly eight ways.">[https://oeis.org/A345770 A345770]</span>||<span title="Numbers that are the sum of six cubes in exactly nine ways.">[https://oeis.org/A345771 A345771]</span>||<span title="Numbers that are the sum of six cubes in exactly ten ways.">[https://oeis.org/A345772 A345772]</span> | ||

+ | |- | ||

+ | | k=7 ||<span title="Numbers that are the sum of seven cubes in exactly one way.">[https://oeis.org/A345773 A345773]</span>||<span title="Numbers that are the sum of seven cubes in exactly two ways.">[https://oeis.org/A345774 A345774]</span>||<span title="Numbers that are the sum of seven cubes in exactly three ways.">[https://oeis.org/A345775 A345775]</span>||<span title="Numbers that are the sum of seven cubes in exactly four ways.">[https://oeis.org/A345776 A345776]</span>||<span title="Numbers that are the sum of seven cubes in exactly five ways.">[https://oeis.org/A345777 A345777]</span>||<span title="Numbers that are the sum of seven cubes in exactly six ways.">[https://oeis.org/A345778 A345778]</span>||<span title="Numbers that are the sum of seven cubes in exactly seven ways.">[https://oeis.org/A345779 A345779]</span>||<span title="Numbers that are the sum of seven cubes in exactly eight ways.">[https://oeis.org/A345780 A345780]</span>||<span title="Numbers that are the sum of seven cubes in exactly nine ways.">[https://oeis.org/A345781 A345781]</span>||<span title="Numbers that are the sum of seven cubes in exactly ten ways.">[https://oeis.org/A345782 A345782]</span> | ||

+ | |- | ||

+ | | k=8 ||<span title="Numbers that are the sum of eight cubes in exactly one way.">[https://oeis.org/A345783 A345783]</span>||<span title="Numbers that are the sum of eight cubes in exactly two ways.">[https://oeis.org/A345784 A345784]</span>||<span title="Numbers that are the sum of eight cubes in exactly three ways.">[https://oeis.org/A345785 A345785]</span>||<span title="Numbers that are the sum of eight cubes in exactly four ways.">[https://oeis.org/A345786 A345786]</span>||<span title="Numbers that are the sum of eight cubes in exactly five ways.">[https://oeis.org/A345787 A345787]</span>||<span title="Numbers that are the sum of eight cubes in exactly six ways.">[https://oeis.org/A345788 A345788]</span>||<span title="Numbers that are the sum of eight cubes in exactly seven ways.">[https://oeis.org/A345789 A345789]</span>||<span title="Numbers that are the sum of eight cubes in exactly eight ways.">[https://oeis.org/A345790 A345790]</span>||<span title="Numbers that are the sum of eight cubes in exactly nine ways.">[https://oeis.org/A345791 A345791]</span>||<span title="Numbers that are the sum of eight cubes in exactly ten ways.">[https://oeis.org/A345792 A345792]</span> | ||

+ | |- | ||

+ | | k=9 ||<span title="Numbers that are the sum of nine cubes in exactly one way.">[https://oeis.org/A345793 A345793]</span>||<span title="Numbers that are the sum of nine cubes in exactly two ways.">[https://oeis.org/A345794 A345794]</span>||<span title="Numbers that are the sum of nine cubes in exactly three ways.">[https://oeis.org/A345795 A345795]</span>||<span title="Numbers that are the sum of nine cubes in exactly four ways.">[https://oeis.org/A345796 A345796]</span>||<span title="Numbers that are the sum of nine cubes in exactly five ways.">[https://oeis.org/A345797 A345797]</span>||<span title="Numbers that are the sum of nine cubes in exactly six ways.">[https://oeis.org/A345798 A345798]</span>||<span title="Numbers that are the sum of nine cubes in exactly seven ways.">[https://oeis.org/A345799 A345799]</span>||<span title="Numbers that are the sum of nine cubes in exactly eight ways.">[https://oeis.org/A345800 A345800]</span>||<span title="Numbers that are the sum of nine cubes in exactly nine ways.">[https://oeis.org/A345801 A345801]</span>||<span title="Numbers that are the sum of nine cubes in exactly ten ways.">[https://oeis.org/A345802 A345802]</span> | ||

+ | |- | ||

+ | | k=10 ||<span title="Numbers that are the sum of ten cubes in exactly one ways.">[https://oeis.org/A345803 A345803]</span>||<span title="Numbers that are the sum of ten cubes in exactly two ways.">[https://oeis.org/A345804 A345804]</span>||<span title="Numbers that are the sum of ten cubes in exactly three ways.">[https://oeis.org/A345805 A345805]</span>||<span title="Numbers that are the sum of ten cubes in exactly four ways.">[https://oeis.org/A345806 A345806]</span>||<span title="Numbers that are the sum of ten cubes in exactly five ways.">[https://oeis.org/A345807 A345807]</span>||<span title="Numbers that are the sum of ten cubes in exactly six ways.">[https://oeis.org/A345808 A345808]</span>||<span title="Numbers that are the sum of ten cubes in exactly seven ways.">[https://oeis.org/A345809 A345809]</span>||<span title="Numbers that are the sum of ten cubes in exactly eight ways.">[https://oeis.org/A345810 A345810]</span>||<span title="Numbers that are the sum of ten cubes in exactly nine ways.">[https://oeis.org/A345811 A345811]</span>||<span title="Numbers that are the sum of ten cubes in exactly ten ways.">[https://oeis.org/A345812 A345812]</span> | ||

+ | |- | ||

+ | |} | ||

+ | ==== Numbers that can be expressed as the sum of k fourth powers in m or more ways (List R4)==== | ||

+ | {| class="wikitable" style="text-align:left" | ||

+ | !   !!m>=1!!m>=2!!m>=3!!m>=4!!m>=5!!m>=6!!m>=7!!m>=8!!m>=9!!m>=10 | ||

+ | |- | ||

+ | | k=1 ||<span title="Fourth powers: a(n) = n^4.">[https://oeis.org/A000583 A000583]</span>|| || || || || || || || ||  | ||

+ | |- | ||

+ | | k=3 || || ||<span title="Numbers that are the sum of three fourth powers in three or more ways.">[https://oeis.org/A344239 A344239]</span>||<span title="Numbers that are the sum of three fourth powers in four or more ways.">[https://oeis.org/A344277 A344277]</span>||<span title="Numbers that are the sum of three fourth powers in five or more ways.">[https://oeis.org/A344364 A344364]</span>||<span title="Numbers that are the sum of three fourth powers in six or more ways.">[https://oeis.org/A344647 A344647]</span>||<span title="Numbers that are the sum of three fourth powers in seven or more ways.">[https://oeis.org/A344729 A344729]</span>||<span title="Numbers that are the sum of three fourth powers in eight or more ways.">[https://oeis.org/A344737 A344737]</span>||<span title="Numbers that are the sum of three fourth powers in nine or more ways.">[https://oeis.org/A344750 A344750]</span>||<span title="Numbers that are the sum of three fourth powers in ten or more ways.">[https://oeis.org/A344862 A344862]</span> | ||

+ | |- | ||

+ | | k=4 || || ||<span title="Numbers that are the sum of four fourth powers in three or more ways.">[https://oeis.org/A344241 A344241]</span>||<span title="Numbers that are the sum of four fourth powers in four or more ways.">[https://oeis.org/A344352 A344352]</span>||<span title="Numbers that are the sum of four fourth powers in five or more ways.">[https://oeis.org/A344356 A344356]</span>||<span title="Numbers that are the sum of four fourth powers in six or more ways.">[https://oeis.org/A344904 A344904]</span>||<span title="Numbers that are the sum of four fourth powers in seven or more ways.">[https://oeis.org/A344922 A344922]</span>||<span title="Numbers that are the sum of four fourth powers in eight or more ways.">[https://oeis.org/A344924 A344924]</span>||<span title="Numbers that are the sum of four fourth powers in nine or more ways.">[https://oeis.org/A344926 A344926]</span>||<span title="Numbers that are the sum of four fourth powers in ten or more ways.">[https://oeis.org/A344928 A344928]</span> | ||

+ | |- | ||

+ | | k=5 || ||<span title="Numbers that are the sum of five fourth powers in two or more ways.">[https://oeis.org/A344238 A344238]</span>||<span title="Numbers that are the sum of five fourth powers in three or more ways.">[https://oeis.org/A344243 A344243]</span>||<span title="Numbers that are the sum of five fourth powers in four or more ways.">[https://oeis.org/A344354 A344354]</span>||<span title="Numbers that are the sum of five fourth powers in five or more ways.">[https://oeis.org/A344358 A344358]</span>||<span title="Numbers that are the sum of five fourth powers in six or more ways.">[https://oeis.org/A344940 A344940]</span>||<span title="Numbers that are the sum of five fourth powers in seven or more ways.">[https://oeis.org/A344942 A344942]</span>||<span title="Numbers that are the sum of five fourth powers in eight or more ways.">[https://oeis.org/A344944 A344944]</span>||<span title="Numbers that are the sum of five fourth powers in nine or more ways.">[https://oeis.org/A341891 A341891]</span>||<span title="Numbers that are the sum of five fourth powers in ten or more ways.">[https://oeis.org/A341897 A341897]</span> | ||

+ | |- | ||

+ | | k=6 || ||<span title="Numbers that are the sum of six fourth powers in two or more ways.">[https://oeis.org/A345559 A345559]</span>||<span title="Numbers that are the sum of six fourth powers in three or more ways.">[https://oeis.org/A345560 A345560]</span>||<span title="Numbers that are the sum of six fourth powers in four or more ways.">[https://oeis.org/A345561 A345561]</span>||<span title="Numbers that are the sum of six fourth powers in five or more ways.">[https://oeis.org/A345562 A345562]</span>||<span title="Numbers that are the sum of six fourth powers in six or more ways.">[https://oeis.org/A345563 A345563]</span>||<span title="Numbers that are the sum of six fourth powers in seven or more ways.">[https://oeis.org/A345564 A345564]</span>||<span title="Numbers that are the sum of six fourth powers in eight or more ways.">[https://oeis.org/A345565 A345565]</span>||<span title="Numbers that are the sum of six fourth powers in nine or more ways.">[https://oeis.org/A345566 A345566]</span>||<span title="Numbers that are the sum of six fourth powers in ten or more ways.">[https://oeis.org/A345567 A345567]</span> | ||

+ | |- | ||

+ | | k=7 || ||<span title="Numbers that are the sum of seven fourth powers in two or more ways.">[https://oeis.org/A345568 A345568]</span>||<span title="Numbers that are the sum of seven fourth powers in three or more ways.">[https://oeis.org/A345569 A345569]</span>||<span title="Numbers that are the sum of seven fourth powers in four or more ways.">[https://oeis.org/A345570 A345570]</span>||<span title="Numbers that are the sum of seven fourth powers in five or more ways.">[https://oeis.org/A345571 A345571]</span>||<span title="Numbers that are the sum of seven fourth powers in six or more ways.">[https://oeis.org/A345572 A345572]</span>||<span title="Numbers that are the sum of seven fourth powers in seven or more ways.">[https://oeis.org/A345573 A345573]</span>||<span title="Numbers that are the sum of seven fourth powers in eight or more ways.">[https://oeis.org/A345574 A345574]</span>||<span title="Numbers that are the sum of seven fourth powers in nine or more ways.">[https://oeis.org/A345575 A345575]</span>||<span title="Numbers that are the sum of seven fourth powers in ten or more ways.">[https://oeis.org/A345576 A345576]</span> | ||

+ | |- | ||

+ | | k=8 || ||<span title="Numbers that are the sum of eight fourth powers in two or more ways.">[https://oeis.org/A345577 A345577]</span>||<span title="Numbers that are the sum of eight fourth powers in three or more ways.">[https://oeis.org/A345578 A345578]</span>||<span title="Numbers that are the sum of eight fourth powers in four or more ways.">[https://oeis.org/A345579 A345579]</span>||<span title="Numbers that are the sum of eight fourth powers in five or more ways.">[https://oeis.org/A345580 A345580]</span>||<span title="Numbers that are the sum of eight fourth powers in six or more ways.">[https://oeis.org/A345581 A345581]</span>||<span title="Numbers that are the sum of eight fourth powers in seven or more ways.">[https://oeis.org/A345582 A345582]</span>||<span title="Numbers that are the sum of eight fourth powers in eight or more ways.">[https://oeis.org/A345583 A345583]</span>||<span title="Numbers that are the sum of eight fourth powers in nine or more ways.">[https://oeis.org/A345584 A345584]</span>||<span title="Numbers that are the sum of eight fourth powers in ten or more ways.">[https://oeis.org/A345585 A345585]</span> | ||

+ | |- | ||

+ | | k=9 || ||<span title="Numbers that are the sum of nine fourth powers in two or more ways.">[https://oeis.org/A345586 A345586]</span>||<span title="Numbers that are the sum of nine fourth powers in three or more ways.">[https://oeis.org/A345587 A345587]</span>||<span title="Numbers that are the sum of nine fourth powers in four or more ways.">[https://oeis.org/A345588 A345588]</span>||<span title="Numbers that are the sum of nine fourth powers in five or more ways.">[https://oeis.org/A345589 A345589]</span>||<span title="Numbers that are the sum of nine fourth powers in six or more ways.">[https://oeis.org/A345590 A345590]</span>||<span title="Numbers that are the sum of nine fourth powers in seven or more ways.">[https://oeis.org/A345591 A345591]</span>||<span title="Numbers that are the sum of nine fourth powers in eight or more ways.">[https://oeis.org/A345592 A345592]</span>||<span title="Numbers that are the sum of nine fourth powers in nine or more ways.">[https://oeis.org/A345593 A345593]</span>||<span title="Numbers that are the sum of nine fourth powers in ten or more ways.">[https://oeis.org/A345594 A345594]</span> | ||

+ | |- | ||

+ | | k=10 || ||<span title="Numbers that are the sum of ten fourth powers in two or more ways.">[https://oeis.org/A345595 A345595]</span>||<span title="Numbers that are the sum of ten fourth powers in three or more ways.">[https://oeis.org/A345596 A345596]</span>||<span title="Numbers that are the sum of ten fourth powers in four or more ways.">[https://oeis.org/A345597 A345597]</span>||<span title="Numbers that are the sum of ten fourth powers in five or more ways.">[https://oeis.org/A345598 A345598]</span>||<span title="Numbers that are the sum of ten fourth powers in six or more ways.">[https://oeis.org/A345599 A345599]</span>||<span title="Numbers that are the sum of ten fourth powers in seven or more ways.">[https://oeis.org/A345600 A345600]</span>||<span title="Numbers that are the sum of ten fourth powers in eight or more ways.">[https://oeis.org/A345601 A345601]</span>||<span title="Numbers that are the sum of ten fourth powers in nine or more ways.">[https://oeis.org/A345602 A345602]</span>||<span title="Numbers that are the sum of ten fourth powers in ten or more ways.">[https://oeis.org/A345603 A345603]</span> | ||

+ | |- | ||

+ | |} | ||

+ | ==== Numbers that can be expressed as the sum of k fourth powers in exactly m ways (List S4)==== | ||

+ | {| class="wikitable" style="text-align:left" | ||

+ | !   !!m=1!!m=2!!m=3!!m=4!!m=5!!m=6!!m=7!!m=8!!m=9!!m=10 | ||

+ | |- | ||

+ | | k=2 ||<span title="Numbers that are the sum of two positive fourth powers in exactly one way.">[https://oeis.org/A344187 A344187]</span>|| || || || || || || || ||  | ||

+ | |- | ||

+ | | k=3 ||<span title="Numbers that are the sum of three fourth powers in exactly one way">[https://oeis.org/A344188 A344188]</span>||<span title="Numbers that are the sum of three fourth powers in exactly two ways.">[https://oeis.org/A344192 A344192]</span>||<span title="Numbers that are the sum of three fourth powers in exactly three ways.">[https://oeis.org/A344240 A344240]</span>||<span title="Numbers that are the sum of three fourth powers in exactly four ways.">[https://oeis.org/A344278 A344278]</span>||<span title="Numbers that are the sum of three fourth powers in exactly five ways.">[https://oeis.org/A344365 A344365]</span>||<span title="Numbers that are the sum of three fourth powers in exactly six ways.">[https://oeis.org/A344648 A344648]</span>||<span title="Numbers that are the sum of three fourth powers in exactly seven ways.">[https://oeis.org/A344730 A344730]</span>||<span title="Numbers that are the sum of three fourth powers in exactly eight ways.">[https://oeis.org/A344738 A344738]</span>||<span title="Numbers that are the sum of three fourth powers in exactly nine ways.">[https://oeis.org/A344751 A344751]</span>||<span title="Numbers that are the sum of three fourth powers in exactly ten ways.">[https://oeis.org/A344861 A344861]</span> | ||

+ | |- | ||

+ | | k=4 ||<span title="Numbers that are the sum of four fourth powers in exactly one way.">[https://oeis.org/A344189 A344189]</span>||<span title="Numbers that are the sum of four fourth powers in exactly two ways">[https://oeis.org/A344193 A344193]</span>||<span title="Numbers that are the sum of four fourth powers in exactly three ways.">[https://oeis.org/A344242 A344242]</span>||<span title="Numbers that are the sum of four fourth powers in exactly four ways.">[https://oeis.org/A344353 A344353]</span>||<span title="Numbers that are the sum of four fourth powers in exactly five ways.">[https://oeis.org/A344357 A344357]</span>||<span title="Numbers that are the sum of four fourth powers in exactly six ways.">[https://oeis.org/A344921 A344921]</span>||<span title="Numbers that are the sum of four fourth powers in exactly seven ways.">[https://oeis.org/A344923 A344923]</span>||<span title="Numbers that are the sum of four fourth powers in exactly eight ways.">[https://oeis.org/A344925 A344925]</span>||<span title="Numbers that are the sum of four fourth powers in exactly nine ways.">[https://oeis.org/A344927 A344927]</span>||<span title="Numbers that are the sum of four fourth powers in exactly ten ways.">[https://oeis.org/A344929 A344929]</span> | ||

+ | |- | ||

+ | | k=5 ||<span title="Numbers that are the sum of five fourth powers in exactly one way.">[https://oeis.org/A344190 A344190]</span>||<span title="Numbers that are the sum of five fourth powers in exactly two ways.">[https://oeis.org/A344237 A344237]</span>||<span title="Numbers that are the sum of five fourth powers in exactly three ways.">[https://oeis.org/A344244 A344244]</span>||<span title="Numbers that are the sum of five fourth powers in exactly four ways.">[https://oeis.org/A344355 A344355]</span>||<span title="Numbers that are the sum of five fourth powers in exactly five ways.">[https://oeis.org/A344359 A344359]</span>||<span title="Numbers that are the sum of five fourth powers in exactly six ways.">[https://oeis.org/A344941 A344941]</span>||<span title="Numbers that are the sum of five fourth powers in exactly seven ways.">[https://oeis.org/A344943 A344943]</span>||<span title="Numbers that are the sum of five fourth powers in exactly eight ways.">[https://oeis.org/A344945 A344945]</span>||<span title="Numbers that are the sum of five fourth powers in exactly nine ways.">[https://oeis.org/A341892 A341892]</span>||<span title="Numbers that are the sum of five fourth powers in exactly ten ways.">[https://oeis.org/A341898 A341898]</span> | ||

+ | |- | ||

+ | | k=6 ||<span title="Numbers that are the sum of six fourth powers in exactly one ways.">[https://oeis.org/A345813 A345813]</span>||<span title="Numbers that are the sum of six fourth powers in exactly two ways.">[https://oeis.org/A345814 A345814]</span>||<span title="Numbers that are the sum of six fourth powers in exactly three ways.">[https://oeis.org/A345815 A345815]</span>||<span title="Numbers that are the sum of six fourth powers in exactly four ways.">[https://oeis.org/A345816 A345816]</span>||<span title="Numbers that are the sum of six fourth powers in exactly five ways.">[https://oeis.org/A345817 A345817]</span>||<span title="Numbers that are the sum of six fourth powers in exactly six ways.">[https://oeis.org/A345818 A345818]</span>||<span title="Numbers that are the sum of six fourth powers in exactly seven ways.">[https://oeis.org/A345819 A345819]</span>||<span title="Numbers that are the sum of six fourth powers in exactly eight ways.">[https://oeis.org/A345820 A345820]</span>||<span title="Numbers that are the sum of six fourth powers in exactly nine ways.">[https://oeis.org/A345821 A345821]</span>||<span title="Numbers that are the sum of six fourth powers in exactly ten ways.">[https://oeis.org/A345822 A345822]</span> | ||

+ | |- | ||

+ | | k=7 ||<span title="Numbers that are the sum of seven fourth powers in exactly one ways.">[https://oeis.org/A345823 A345823]</span>||<span title="Numbers that are the sum of seven fourth powers in exactly two ways.">[https://oeis.org/A345824 A345824]</span>||<span title="Numbers that are the sum of seven fourth powers in exactly three ways.">[https://oeis.org/A345825 A345825]</span>||<span title="Numbers that are the sum of seven fourth powers in exactly four ways.">[https://oeis.org/A345826 A345826]</span>||<span title="Numbers that are the sum of seven fourth powers in exactly five ways.">[https://oeis.org/A345827 A345827]</span>||<span title="Numbers that are the sum of seven fourth powers in exactly six ways.">[https://oeis.org/A345828 A345828]</span>||<span title="Numbers that are the sum of seven fourth powers in exactly seven ways.">[https://oeis.org/A345829 A345829]</span>||<span title="Numbers that are the sum of seven fourth powers in exactly eight ways.">[https://oeis.org/A345830 A345830]</span>||<span title="Numbers that are the sum of seven fourth powers in exactly nine ways.">[https://oeis.org/A345831 A345831]</span>||<span title="Numbers that are the sum of seven fourth powers in exactly ten ways.">[https://oeis.org/A345832 A345832]</span> | ||

+ | |- | ||

+ | | k=8 ||<span title="Numbers that are the sum of eight fourth powers in exactly one ways.">[https://oeis.org/A345833 A345833]</span>||<span title="Numbers that are the sum of eight fourth powers in exactly two ways.">[https://oeis.org/A345834 A345834]</span>||<span title="Numbers that are the sum of eight fourth powers in exactly three ways.">[https://oeis.org/A345835 A345835]</span>||<span title="Numbers that are the sum of eight fourth powers in exactly four ways.">[https://oeis.org/A345836 A345836]</span>||<span title="Numbers that are the sum of eight fourth powers in exactly five ways.">[https://oeis.org/A345837 A345837]</span>||<span title="Numbers that are the sum of eight fourth powers in exactly six ways.">[https://oeis.org/A345838 A345838]</span>||<span title="Numbers that are the sum of eight fourth powers in exactly seven ways.">[https://oeis.org/A345839 A345839]</span>||<span title="Numbers that are the sum of eight fourth powers in exactly eight ways.">[https://oeis.org/A345840 A345840]</span>||<span title="Numbers that are the sum of eight fourth powers in exactly nine ways.">[https://oeis.org/A345841 A345841]</span>||<span title="Numbers that are the sum of eight fourth powers in exactly ten ways.">[https://oeis.org/A345842 A345842]</span> | ||

+ | |- | ||

+ | | k=9 ||<span title="Numbers that are the sum of nine fourth powers in exactly one ways.">[https://oeis.org/A345843 A345843]</span>||<span title="Numbers that are the sum of nine fourth powers in exactly two ways.">[https://oeis.org/A345844 A345844]</span>||<span title="Numbers that are the sum of nine fourth powers in exactly three ways.">[https://oeis.org/A345845 A345845]</span>||<span title="Numbers that are the sum of nine fourth powers in exactly four ways.">[https://oeis.org/A345846 A345846]</span>||<span title="Numbers that are the sum of nine fourth powers in exactly five ways.">[https://oeis.org/A345847 A345847]</span>||<span title="Numbers that are the sum of nine fourth powers in exactly six ways.">[https://oeis.org/A345848 A345848]</span>||<span title="Numbers that are the sum of nine fourth powers in exactly seven ways.">[https://oeis.org/A345849 A345849]</span>||<span title="Numbers that are the sum of nine fourth powers in exactly eight ways.">[https://oeis.org/A345850 A345850]</span>||<span title="Numbers that are the sum of nine fourth powers in exactly nine ways.">[https://oeis.org/A345851 A345851]</span>||<span title="Numbers that are the sum of nine fourth powers in exactly ten ways.">[https://oeis.org/A345852 A345852]</span> | ||

+ | |- | ||

+ | | k=10 ||<span title="Numbers that are the sum of ten fourth powers in exactly one ways.">[https://oeis.org/A345853 A345853]</span>||<span title="Numbers that are the sum of ten fourth powers in exactly two ways.">[https://oeis.org/A345854 A345854]</span>||<span title="Numbers that are the sum of ten fourth powers in exactly three ways.">[https://oeis.org/A345855 A345855]</span>||<span title="Numbers that are the sum of ten fourth powers in exactly four ways.">[https://oeis.org/A345856 A345856]</span>||<span title="Numbers that are the sum of ten fourth powers in exactly five ways.">[https://oeis.org/A345857 A345857]</span>||<span title="Numbers that are the sum of ten fourth powers in exactly six ways.">[https://oeis.org/A345858 A345858]</span>||<span title="Numbers that are the sum of ten fourth powers in exactly seven ways.">[https://oeis.org/A345859 A345859]</span>||<span title="Numbers that are the sum of ten fourth powers in exactly eight ways.">[https://oeis.org/A345860 A345860]</span>||<span title="Numbers that are the sum of ten fourth powers in exactly nine ways.">[https://oeis.org/A345861 A345861]</span>||<span title="Numbers that are the sum of ten fourth powers in exactly ten ways.">[https://oeis.org/A345862 A345862]</span> | ||

+ | |- | ||

+ | |} | ||

+ | ==== Numbers that can be expressed as the sum of k fifth powers in m or more ways (List R5)==== | ||

+ | {| class="wikitable" style="text-align:left" | ||

+ | !   !!m>=1!!m>=2!!m>=3!!m>=4!!m>=5!!m>=6!!m>=7!!m>=8!!m>=9!!m>=10 | ||

+ | |- | ||

+ | | k=1 ||<span title="Fifth powers: a(n) = n^5.">[https://oeis.org/A000584 A000584]</span>|| || || || || || || || ||  | ||

+ | |- | ||

+ | | k=3 || ||<span title="Numbers that are the sum of three fifth powers in two or more ways.">[https://oeis.org/A345010 A345010]</span>|| || || || || || || ||  | ||

+ | |- | ||

+ | | k=4 || ||<span title="Numbers that are the sum of four fifth powers in two or more ways.">[https://oeis.org/A344644 A344644]</span>||<span title="Numbers that are the sum of four fifth powers in three or more ways.">[https://oeis.org/A345337 A345337]</span>|| || || || || || ||  | ||

+ | |- | ||

+ | | k=5 || ||<span title="Numbers that are the sum of five fifth powers in two or more ways.">[https://oeis.org/A342685 A342685]</span>||<span title="Numbers that are the sum of five fifth powers in three or more ways.">[https://oeis.org/A342687 A342687]</span>||<span title="Numbers that are the sum of five fifth powers in four or more ways.">[https://oeis.org/A344518 A344518]</span>||<span title="Numbers that are the sum of five fifth powers in five or more ways.">[https://oeis.org/A345863 A345863]</span>||<span title="Numbers that are the sum of five fifth powers in six or more ways.">[https://oeis.org/A345864 A345864]</span>|| || || ||  | ||

+ | |- | ||

+ | | k=6 || ||<span title="Numbers that are the sum of six fifth powers in two or more ways.">[https://oeis.org/A345507 A345507]</span>||<span title="Numbers that are the sum of six fifth powers in three or more ways.">[https://oeis.org/A345604 A345604]</span>||<span title="Numbers that are the sum of six fifth powers in four or more ways.">[https://oeis.org/A345718 A345718]</span>||<span title="Numbers that are the sum of six fifth powers in five or more ways.">[https://oeis.org/A345719 A345719]</span>||<span title="Numbers that are the sum of six fifth powers in six or more ways.">[https://oeis.org/A345720 A345720]</span>||<span title="Numbers that are the sum of six fifth powers in seven or more ways.">[https://oeis.org/A345721 A345721]</span>||<span title="Numbers that are the sum of six fifth powers in eight or more ways.">[https://oeis.org/A345722 A345722]</span>||<span title="Numbers that are the sum of six fifth powers in nine or more ways.">[https://oeis.org/A345723 A345723]</span>||<span title="Numbers that are the sum of six fifth powers in ten or more ways.">[https://oeis.org/A344196 A344196]</span> | ||

+ | |- | ||

+ | | k=7 || ||<span title="Numbers that are the sum of seven fifth powers in two or more ways.">[https://oeis.org/A345605 A345605]</span>||<span title="Numbers that are the sum of seven fifth powers in three or more ways.">[https://oeis.org/A345606 A345606]</span>||<span title="Numbers that are the sum of seven fifth powers in four or more ways.">[https://oeis.org/A345607 A345607]</span>||<span title="Numbers that are the sum of seven fifth powers in five or more ways.">[https://oeis.org/A345608 A345608]</span>||<span title="Numbers that are the sum of seven fifth powers in six or more ways.">[https://oeis.org/A345609 A345609]</span>||<span title="Numbers that are the sum of seven fifth powers in seven or more ways.">[https://oeis.org/A345629 A345629]</span>||<span title="Numbers that are the sum of seven fifth powers in eight or more ways.">[https://oeis.org/A345630 A345630]</span>||<span title="Numbers that are the sum of seven fifth powers in nine or more ways.">[https://oeis.org/A345631 A345631]</span>||<span title="Numbers that are the sum of seven fifth powers in ten or more ways.">[https://oeis.org/A345643 A345643]</span> | ||

+ | |- | ||

+ | | k=8 || ||<span title="Numbers that are the sum of eight fifth powers in two or more ways.">[https://oeis.org/A345610 A345610]</span>||<span title="Numbers that are the sum of eight fifth powers in three or more ways.">[https://oeis.org/A345611 A345611]</span>||<span title="Numbers that are the sum of eight fifth powers in four or more ways.">[https://oeis.org/A345612 A345612]</span>||<span title="Numbers that are the sum of eight fifth powers in five or more ways.">[https://oeis.org/A345613 A345613]</span>||<span title="Numbers that are the sum of eight fifth powers in six or more ways.">[https://oeis.org/A345614 A345614]</span>||<span title="Numbers that are the sum of eight fifth powers in seven or more ways.">[https://oeis.org/A345615 A345615]</span>||<span title="Numbers that are the sum of eight fifth powers in eight or more ways.">[https://oeis.org/A345616 A345616]</span>||<span title="Numbers that are the sum of eight fifth powers in nine or more ways.">[https://oeis.org/A345617 A345617]</span>||<span title="Numbers that are the sum of eight fifth powers in ten or more ways.">[https://oeis.org/A345618 A345618]</span> | ||

+ | |- | ||

+ | | k=9 || ||<span title="Numbers that are the sum of nine fifth powers in two or more ways.">[https://oeis.org/A345619 A345619]</span>||<span title="Numbers that are the sum of nine fifth powers in three or more ways.">[https://oeis.org/A345620 A345620]</span>||<span title="Numbers that are the sum of nine fifth powers in four or more ways.">[https://oeis.org/A345621 A345621]</span>||<span title="Numbers that are the sum of nine fifth powers in five or more ways.">[https://oeis.org/A345622 A345622]</span>||<span title="Numbers that are the sum of nine fifth powers in six or more ways.">[https://oeis.org/A345623 A345623]</span>||<span title="Numbers that are the sum of nine fifth powers in seven or more ways.">[https://oeis.org/A345624 A345624]</span>||<span title="Numbers that are the sum of nine fifth powers in eight or more ways.">[https://oeis.org/A345625 A345625]</span>||<span title="Numbers that are the sum of nine fifth powers in nine or more ways.">[https://oeis.org/A345626 A345626]</span>||<span title="Numbers that are the sum of nine fifth powers in ten or more ways.">[https://oeis.org/A345627 A345627]</span> | ||

+ | |- | ||

+ | | k=10 || ||<span title="Numbers that are the sum of ten fifth powers in two or more ways.">[https://oeis.org/A345634 A345634]</span>||<span title="Numbers that are the sum of ten fifth powers in three or more ways.">[https://oeis.org/A345635 A345635]</span>||<span title="Numbers that are the sum of ten fifth powers in four or more ways.">[https://oeis.org/A345636 A345636]</span>||<span title="Numbers that are the sum of ten fifth powers in five or more ways.">[https://oeis.org/A345637 A345637]</span>||<span title="Numbers that are the sum of ten fifth powers in six or more ways.">[https://oeis.org/A345638 A345638]</span>||<span title="Numbers that are the sum of ten fifth powers in seven or more ways.">[https://oeis.org/A345639 A345639]</span>||<span title="Numbers that are the sum of ten fifth powers in eight or more ways.">[https://oeis.org/A345640 A345640]</span>||<span title="Numbers that are the sum of ten fifth powers in nine or more ways.">[https://oeis.org/A345641 A345641]</span>||<span title="Numbers that are the sum of ten fifth powers in ten or more ways.">[https://oeis.org/A345642 A345642]</span> | ||

+ | |- | ||

+ | |} | ||

+ | ==== Numbers that can be expressed as the sum of k fifth powers in exactly m ways (List S5)==== | ||

+ | {| class="wikitable" style="text-align:left" | ||

+ | !   !!m=1!!m=2!!m=3!!m=4!!m=5!!m=6!!m=7!!m=8!!m=9!!m=10 | ||

|- | |- | ||

− | | | + | | k=3 ||<span title="Numbers that are the sum of three fifth powers in exactly one way.">[https://oeis.org/A344641 A344641]</span>|| || || || || || || || ||  |

|- | |- | ||

− | | | + | | k=4 ||<span title="Numbers that are the sum of four fifth powers in exactly one way.">[https://oeis.org/A344642 A344642]</span>||<span title="Numbers that are the sum of four fifth powers in exactly two ways.">[https://oeis.org/A344645 A344645]</span>|| || || || || || || ||  |

|- | |- | ||

− | | | + | | k=5 ||<span title="Numbers that are the sum of five fifth powers in exactly one way.">[https://oeis.org/A344643 A344643]</span>||<span title="Numbers that are the sum of five fifth powers in exactly two ways.">[https://oeis.org/A342686 A342686]</span>||<span title="Numbers that are the sum of five fifth powers in exactly three ways.">[https://oeis.org/A342688 A342688]</span>||<span title="Numbers that are the sum of five fifth powers in exactly four ways.">[https://oeis.org/A344519 A344519]</span>||<span title="Numbers that are the sum of five fifth powers in exactly five ways.">[https://oeis.org/A346257 A346257]</span>|| || || || ||  |

|- | |- | ||

− | | | + | | k=6 ||<span title="Numbers that are the sum of six fifth powers in exactly one way.">[https://oeis.org/A346356 A346356]</span>||<span title="Numbers that are the sum of six fifth powers in exactly two ways.">[https://oeis.org/A346357 A346357]</span>||<span title="Numbers that are the sum of six fifth powers in exactly three ways.">[https://oeis.org/A346358 A346358]</span>||<span title="Numbers that are the sum of six fifth powers in exactly four ways.">[https://oeis.org/A346359 A346359]</span>||<span title="Numbers that are the sum of six fifth powers in exactly five ways.">[https://oeis.org/A346360 A346360]</span>||<span title="Numbers that are the sum of six fifth powers in exactly six ways.">[https://oeis.org/A346361 A346361]</span>||<span title="Numbers that are the sum of six fifth powers in exactly seven ways.">[https://oeis.org/A346362 A346362]</span>||<span title="Numbers that are the sum of six fifth powers in exactly eight ways.">[https://oeis.org/A346363 A346363]</span>||<span title="Numbers that are the sum of six fifth powers in exactly nine ways.">[https://oeis.org/A346364 A346364]</span>||<span title="Numbers that are the sum of six fifth powers in exactly ten ways.">[https://oeis.org/A346365 A346365]</span> |

|- | |- | ||

− | | | + | | k=7 ||<span title="Numbers that are the sum of seven fifth powers in exactly one way.">[https://oeis.org/A346278 A346278]</span>||<span title="Numbers that are the sum of seven fifth powers in exactly two ways.">[https://oeis.org/A346279 A346279]</span>||<span title="Numbers that are the sum of seven fifth powers in exactly three ways.">[https://oeis.org/A346280 A346280]</span>||<span title="Numbers that are the sum of seven fifth powers in exactly four ways.">[https://oeis.org/A346281 A346281]</span>||<span title="Numbers that are the sum of seven fifth powers in exactly five ways.">[https://oeis.org/A346282 A346282]</span>||<span title="Numbers that are the sum of seven fifth powers in exactly six ways.">[https://oeis.org/A346283 A346283]</span>||<span title="Numbers that are the sum of seven fifth powers in exactly seven ways.">[https://oeis.org/A346284 A346284]</span>||<span title="Numbers that are the sum of seven fifth powers in exactly eight ways.">[https://oeis.org/A346285 A346285]</span>||<span title="Numbers that are the sum of seven fifth powers in exactly nine ways.">[https://oeis.org/A346286 A346286]</span>||<span title="Numbers that are the sum of seven fifth powers in exactly ten ways.">[https://oeis.org/A346259 A346259]</span> |

|- | |- | ||

− | | | + | | k=8 ||<span title="Numbers that are the sum of eight fifth powers in exactly one way.">[https://oeis.org/A346326 A346326]</span>||<span title="Numbers that are the sum of eight fifth powers in exactly two ways.">[https://oeis.org/A346327 A346327]</span>||<span title="Numbers that are the sum of eight fifth powers in exactly three ways.">[https://oeis.org/A346328 A346328]</span>||<span title="Numbers that are the sum of eight fifth powers in exactly four ways.">[https://oeis.org/A346329 A346329]</span>||<span title="Numbers that are the sum of eight fifth powers in exactly five ways.">[https://oeis.org/A346330 A346330]</span>||<span title="Numbers that are the sum of eight fifth powers in exactly six ways.">[https://oeis.org/A346331 A346331]</span>||<span title="Numbers that are the sum of eight fifth powers in exactly seven ways.">[https://oeis.org/A346332 A346332]</span>||<span title="Numbers that are the sum of eight fifth powers in exactly eight ways.">[https://oeis.org/A346333 A346333]</span>||<span title="Numbers that are the sum of eight fifth powers in exactly nine ways.">[https://oeis.org/A346334 A346334]</span>||<span title="Numbers that are the sum of eight fifth powers in exactly ten ways.">[https://oeis.org/A346335 A346335]</span> |

|- | |- | ||

− | | | + | | k=9 ||<span title="Numbers that are the sum of nine fifth powers in exactly one way.">[https://oeis.org/A346336 A346336]</span>||<span title="Numbers that are the sum of nine fifth powers in exactly two ways.">[https://oeis.org/A346337 A346337]</span>||<span title="Numbers that are the sum of nine fifth powers in exactly three ways.">[https://oeis.org/A346338 A346338]</span>||<span title="Numbers that are the sum of nine fifth powers in exactly four ways.">[https://oeis.org/A346339 A346339]</span>||<span title="Numbers that are the sum of nine fifth powers in exactly five ways.">[https://oeis.org/A346340 A346340]</span>||<span title="Numbers that are the sum of nine fifth powers in exactly six ways.">[https://oeis.org/A346341 A346341]</span>||<span title="Numbers that are the sum of nine fifth powers in exactly seven ways.">[https://oeis.org/A346342 A346342]</span>||<span title="Numbers that are the sum of nine fifth powers in exactly eight ways.">[https://oeis.org/A346343 A346343]</span>||<span title="Numbers that are the sum of nine fifth powers in exactly nine ways.">[https://oeis.org/A346344 A346344]</span>||<span title="Numbers that are the sum of nine fifth powers in exactly ten ways.">[https://oeis.org/A346345 A346345]</span> |

|- | |- | ||

− | | | + | | k=10 ||<span title="Numbers that are the sum of ten fifth powers in exactly one way.">[https://oeis.org/A346346 A346346]</span>||<span title="Numbers that are the sum of ten fifth powers in exactly two ways.">[https://oeis.org/A346347 A346347]</span>||<span title="Numbers that are the sum of ten fifth powers in exactly three ways.">[https://oeis.org/A346348 A346348]</span>||<span title="Numbers that are the sum of ten fifth powers in exactly four ways.">[https://oeis.org/A346349 A346349]</span>||<span title="Numbers that are the sum of ten fifth powers in exactly five ways.">[https://oeis.org/A346350 A346350]</span>||<span title="Numbers that are the sum of ten fifth powers in exactly six ways.">[https://oeis.org/A346351 A346351]</span>||<span title="Numbers that are the sum of ten fifth powers in exactly seven ways.">[https://oeis.org/A346352 A346352]</span>||<span title="Numbers that are the sum of ten fifth powers in exactly eight ways.">[https://oeis.org/A346353 A346353]</span>||<span title="Numbers that are the sum of ten fifth powers in exactly nine ways.">[https://oeis.org/A346354 A346354]</span>||<span title="Numbers that are the sum of ten fifth powers in exactly ten ways.">[https://oeis.org/A346355 A346355]</span> |

|- | |- | ||

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## Revision as of 19:40, 19 August 2021

## Contents

- 1 Sums of like powers
- 2 New Lists
- 2.1 Numbers that can be expressed as the sum of k squares in m or more ways (List R2)
- 2.2 Numbers that can be expressed as the sum of k squares in exactly m ways (List S2)
- 2.3 Numbers that can be expressed as the sum of k cubes in m or more ways (List R3)
- 2.4 Numbers that can be expressed as the sum of k cubes in exactly m ways (List S3)
- 2.5 Numbers that can be expressed as the sum of k fourth powers in m or more ways (List R4)
- 2.6 Numbers that can be expressed as the sum of k fourth powers in exactly m ways (List S4)
- 2.7 Numbers that can be expressed as the sum of k fifth powers in m or more ways (List R5)
- 2.8 Numbers that can be expressed as the sum of k fifth powers in exactly m ways (List S5)

### Sums of like powers

#### Sums of k m-th powers >= 0 (List A)

m=2 | m=3 | m=4 | m=5 | m=6 | m=7 | m=8 | m=9 | m=10 | m=13 | |||
---|---|---|---|---|---|---|---|---|---|---|---|---|

k>=2 | A176209 | |||||||||||

k=-1 | A336448 | A294287 | A294288 | A294300 | A294301 | A294302 | A155468 | A007487 | A294305 | A181134 | ||

k=2 | A140328 | A004999 | A018786 | |||||||||

k=3 | A294713 | A332201 | A193244 | |||||||||

k=4 | A001245 |

#### Sums of exactly k positive m-th powers > 0 (List B)

m=2 | m=3 | m=4 | m=5 | m=6 | m=7 | m=8 | m=9 | m=10 | m=11 | |
---|---|---|---|---|---|---|---|---|---|---|

k=2 | A024509 | A003325 | A003336 | A003347 | A003358 | A003369 | A003380 | A003391 | A004802 | A004813 |

k=3 | A024795 | A024981 | A309762 | A003348 | A003359 | A003370 | A003381 | A003392 | A004803 | A004814 |

k=4 | A000414 | A309763 | A003349 | A003360 | A003371 | A003382 | A003393 | A004804 | A004815 | |

k=5 | A047700 | A003328 | A003339 | A003350 | A003361 | A003372 | A003383 | A003394 | A004805 | A004816 |

k=6 | A003329 | A003340 | A003351 | A003362 | A003373 | A003384 | A003395 | A004806 | A004817 | |

k=7 | A003330 | A003341 | A003352 | A003363 | A003374 | A003385 | A003396 | A004807 | A004818 | |

k=8 | A003331 | A003342 | A003353 | A003364 | A003375 | A003386 | A003397 | A004808 | A004819 | |

k=9 | A003332 | A003343 | A003354 | A003365 | A003376 | A003387 | A003398 | A004809 | A004820 | |

k=10 | A003333 | A003344 | A003355 | A003366 | A003377 | A003388 | A003399 | A004810 | A004821 | |

k=11 | A003334 | A003345 | A003356 | A003367 | A003378 | A003389 | A004800 | A004811 | A004822 | |

k=12 | A003335 | A003346 | A003357 | A003368 | A003379 | A003390 | A004801 | A004812 | A004823 | |

k=13 | A047724 | A123294 | ||||||||

k=14 | A047725 | A123295 |

#### Sums of at most k positive m-th powers > 0 (List C)

m=3 | m=4 | m=5 | m=6 | m=7 | m=8 | m=9 | m=10 | m=11 | |
---|---|---|---|---|---|---|---|---|---|

k<=2 | A004831 | A004842 | A004853 | A004864 | A004875 | A004886 | A004897 | A004908 | |

k<=3 | A004825 | A004832 | A004843 | A004854 | A004865 | A004876 | A004887 | A004898 | A004909 |

k<=4 | A004826 | A004833 | A004844 | A004855 | A004866 | A004877 | A004888 | A004899 | A004910 |

k<=5 | A004827 | A004834 | A004845 | A004856 | A004867 | A004878 | A004889 | A004900 | A004911 |

k<=6 | A004828 | A004835 | A004846 | A004857 | A004868 | A004879 | A004890 | A004901 | A004912 |

k<=7 | A004829 | A004836 | A004847 | A004858 | A004869 | A004880 | A004891 | A004902 | A004913 |

k<=8 | A004830 | A004837 | A004848 | A004859 | A004870 | A004881 | A004892 | A004903 | A004914 |

k<=9 | A004838 | A004849 | A004860 | A004871 | A004882 | A004893 | A004904 | A004915 | |

k<=10 | A004839 | A004850 | A004861 | A004872 | A004883 | A004894 | A004905 | A004916 | |

k<=11 | A004840 | A004851 | A004862 | A004873 | A004884 | A004895 | A004906 | A004917 | |

k<=12 | A004841 | A004852 | A004863 | A004874 | A004885 | A004896 | A004907 | A004918 |

#### Sums of k positive m-th powers > 1 (List D)

m=2 | m=3 | |
---|---|---|

k=-1 | A078131 | |

k=2 | A294073 | |

k=3 | A302359 | A302360 |

#### Numbers that have exactly k representations as the sum of m squares >= 0 (List E)

m=2 | m=5 | m=6 | m=7 | |||
---|---|---|---|---|---|---|

k=1 | A295484 | |||||

k=2 | A085625 | A295150 | A295485 | A295742 | ||

k=3 | A000443 | A295151 | A295486 | A295743 | ||

k=4 | A295152 | A295487 | A295744 | |||

k=5 | A294716 | A295153 | A295488 | A295745 | ||

k=6 | A295154 | A295489 | A295747 | |||

k=7 | A295155 | A295490 | A295748 | |||

k=8 | A295156 | A295491 | A295749 | |||

k=9 | A295157 | A295492 | A295750 | |||

k=10 | A295158 | A295493 | A295751 |

### New Lists

#### Numbers that can be expressed as the sum of k squares in m or more ways (List R2)

m>=1 | m>=2 | m>=3 | m>=4 | m>=5 | m>=6 | m>=7 | m>=8 | m>=9 | m>=10 | |
---|---|---|---|---|---|---|---|---|---|---|

k=1 | A000290 | |||||||||

k=2 | A007692 | A025313 | A025314 | A025315 | A025316 | A025317 | A025318 | A025319 | A025320 | |

k=3 | A024796 | A024804 | A025349 | A025350 | A025351 | A025352 | A025353 | A025354 | A025355 | A025356 |

k=4 | A259058 | A025387 | A025388 | A025389 | A025390 | A025391 | A025392 | A025393 | A025394 | |

k=5 | A344795 | A344796 | A344797 | A344798 | A344799 | A344800 | A344801 | A344802 | A344803 | |

k=6 | A344805 | A344806 | A344807 | A344808 | A344809 | A344810 | A344811 | A344812 | A345476 | A345477 |

k=7 | A345478 | A345479 | A345480 | A345481 | A345482 | A345483 | A345484 | A345485 | A345486 | A345487 |

k=8 | A345488 | A345489 | A345490 | A345491 | A345492 | A345493 | A345494 | A345495 | A345496 | A345497 |

k=9 | A345498 | A345499 | A345500 | A345501 | A345502 | A345503 | A345504 | A345505 | A346803 | |

k=10 | A345508 | A345509 | A345510 | A346804 | A346805 | A346806 | A346807 | A346808 |

#### Numbers that can be expressed as the sum of k squares in exactly m ways (List S2)

m=1 | m=2 | m=3 | m=4 | m=5 | m=6 | m=7 | m=8 | m=9 | m=10 | m=11 | |
---|---|---|---|---|---|---|---|---|---|---|---|

k=2 | A025302 | A085625 | A025304 | A025305 | A294716 | A025307 | A025308 | A025309 | A025310 | A025311 | A236711 |

k=3 | A025339 | A025340 | A025341 | A025342 | A025343 | A025344 | A025345 | A025346 | A025347 | A025348 | |

k=4 | A025376 | A025377 | A025378 | A025379 | A025380 | A025381 | A025382 | A025383 | A025384 | A025385 | |

k=5 | A294675 | A295150 | A295151 | A295152 | A295153 | A295154 | A295155 | A295156 | A295157 | A295158 | |

k=6 | A295670 | A295692 | A295693 | A295694 | A295695 | A295696 | A295697 | A295698 | A295699 | A295700 | |

k=7 | A295797 | A295799 | A295800 | A295801 | A295802 | A295803 | A295804 | A295805 | A295806 | A295807 |

#### Numbers that can be expressed as the sum of k cubes in m or more ways (List R3)

m>=1 | m>=2 | m>=3 | m>=4 | m>=5 | m>=6 | m>=7 | m>=8 | m>=9 | m>=10 | |
---|---|---|---|---|---|---|---|---|---|---|

k=1 | A000578 | |||||||||

k=2 | A001235 | A018787 | A023051 | A051167 | ||||||

k=3 | A024974 | A025402 | A343968 | A343967 | A345083 | A345086 | A345087 | A345119 | A345121 | |

k=4 | A003327 | A259060 | A025413 | A343971 | A343987 | A345148 | A345150 | A345152 | A345146 | A345155 |

k=5 | A343702 | A343704 | A344034 | A343989 | A345174 | A345180 | A345183 | A345185 | A345187 | |

k=6 | A345511 | A345512 | A345513 | A345514 | A345515 | A345516 | A345517 | A345518 | A345519 | |

k=7 | A345520 | A345521 | A345522 | A345523 | A345524 | A345525 | A345526 | A345527 | A345506 | |

k=8 | A345532 | A345533 | A345534 | A345535 | A345536 | A345537 | A345538 | A345539 | A345540 | |

k=9 | A345541 | A345542 | A345543 | A345544 | A345545 | A345546 | A345547 | A345548 | A345549 | |

k=10 | A345550 | A345551 | A345552 | A345553 | A345554 | A345555 | A345556 | A345557 | A345558 |

#### Numbers that can be expressed as the sum of k cubes in exactly m ways (List S3)

m=1 | m=2 | m=3 | m=4 | m=5 | m=6 | m=7 | m=8 | m=9 | m=10 | |
---|---|---|---|---|---|---|---|---|---|---|

k=2 | A338667 | A343708 | A344804 | A345865 | ||||||

k=3 | A025399 | A025400 | A025401 | A343969 | A343970 | A345084 | A345085 | A345088 | A345120 | A345122 |

k=4 | A025408 | A025409 | A025410 | A343972 | A343988 | A345149 | A345151 | A345153 | A345154 | A345156 |

k=5 | A048926 | A048927 | A343705 | A344035 | A345175 | A345181 | A345184 | A345186 | A345188 | |

k=6 | A048929 | A048930 | A048931 | A345766 | A345767 | A345768 | A345769 | A345770 | A345771 | A345772 |

k=7 | A345773 | A345774 | A345775 | A345776 | A345777 | A345778 | A345779 | A345780 | A345781 | A345782 |

k=8 | A345783 | A345784 | A345785 | A345786 | A345787 | A345788 | A345789 | A345790 | A345791 | A345792 |

k=9 | A345793 | A345794 | A345795 | A345796 | A345797 | A345798 | A345799 | A345800 | A345801 | A345802 |

k=10 | A345803 | A345804 | A345805 | A345806 | A345807 | A345808 | A345809 | A345810 | A345811 | A345812 |

#### Numbers that can be expressed as the sum of k fourth powers in m or more ways (List R4)

m>=1 | m>=2 | m>=3 | m>=4 | m>=5 | m>=6 | m>=7 | m>=8 | m>=9 | m>=10 | |
---|---|---|---|---|---|---|---|---|---|---|

k=1 | A000583 | |||||||||

k=3 | A344239 | A344277 | A344364 | A344647 | A344729 | A344737 | A344750 | A344862 | ||

k=4 | A344241 | A344352 | A344356 | A344904 | A344922 | A344924 | A344926 | A344928 | ||

k=5 | A344238 | A344243 | A344354 | A344358 | A344940 | A344942 | A344944 | A341891 | A341897 | |

k=6 | A345559 | A345560 | A345561 | A345562 | A345563 | A345564 | A345565 | A345566 | A345567 | |

k=7 | A345568 | A345569 | A345570 | A345571 | A345572 | A345573 | A345574 | A345575 | A345576 | |

k=8 | A345577 | A345578 | A345579 | A345580 | A345581 | A345582 | A345583 | A345584 | A345585 | |

k=9 | A345586 | A345587 | A345588 | A345589 | A345590 | A345591 | A345592 | A345593 | A345594 | |

k=10 | A345595 | A345596 | A345597 | A345598 | A345599 | A345600 | A345601 | A345602 | A345603 |

#### Numbers that can be expressed as the sum of k fourth powers in exactly m ways (List S4)

m=1 | m=2 | m=3 | m=4 | m=5 | m=6 | m=7 | m=8 | m=9 | m=10 | |
---|---|---|---|---|---|---|---|---|---|---|

k=2 | A344187 | |||||||||

k=3 | A344188 | A344192 | A344240 | A344278 | A344365 | A344648 | A344730 | A344738 | A344751 | A344861 |

k=4 | A344189 | A344193 | A344242 | A344353 | A344357 | A344921 | A344923 | A344925 | A344927 | A344929 |

k=5 | A344190 | A344237 | A344244 | A344355 | A344359 | A344941 | A344943 | A344945 | A341892 | A341898 |

k=6 | A345813 | A345814 | A345815 | A345816 | A345817 | A345818 | A345819 | A345820 | A345821 | A345822 |

k=7 | A345823 | A345824 | A345825 | A345826 | A345827 | A345828 | A345829 | A345830 | A345831 | A345832 |

k=8 | A345833 | A345834 | A345835 | A345836 | A345837 | A345838 | A345839 | A345840 | A345841 | A345842 |

k=9 | A345843 | A345844 | A345845 | A345846 | A345847 | A345848 | A345849 | A345850 | A345851 | A345852 |

k=10 | A345853 | A345854 | A345855 | A345856 | A345857 | A345858 | A345859 | A345860 | A345861 | A345862 |

#### Numbers that can be expressed as the sum of k fifth powers in m or more ways (List R5)

m>=1 | m>=2 | m>=3 | m>=4 | m>=5 | m>=6 | m>=7 | m>=8 | m>=9 | m>=10 | |
---|---|---|---|---|---|---|---|---|---|---|

k=1 | A000584 | |||||||||

k=3 | A345010 | |||||||||

k=4 | A344644 | A345337 | ||||||||

k=5 | A342685 | A342687 | A344518 | A345863 | A345864 | |||||

k=6 | A345507 | A345604 | A345718 | A345719 | A345720 | A345721 | A345722 | A345723 | A344196 | |

k=7 | A345605 | A345606 | A345607 | A345608 | A345609 | A345629 | A345630 | A345631 | A345643 | |

k=8 | A345610 | A345611 | A345612 | A345613 | A345614 | A345615 | A345616 | A345617 | A345618 | |

k=9 | A345619 | A345620 | A345621 | A345622 | A345623 | A345624 | A345625 | A345626 | A345627 | |

k=10 | A345634 | A345635 | A345636 | A345637 | A345638 | A345639 | A345640 | A345641 | A345642 |

#### Numbers that can be expressed as the sum of k fifth powers in exactly m ways (List S5)

m=1 | m=2 | m=3 | m=4 | m=5 | m=6 | m=7 | m=8 | m=9 | m=10 | |
---|---|---|---|---|---|---|---|---|---|---|

k=3 | A344641 | |||||||||

k=4 | A344642 | A344645 | ||||||||

k=5 | A344643 | A342686 | A342688 | A344519 | A346257 | |||||

k=6 | A346356 | A346357 | A346358 | A346359 | A346360 | A346361 | A346362 | A346363 | A346364 | A346365 |

k=7 | A346278 | A346279 | A346280 | A346281 | A346282 | A346283 | A346284 | A346285 | A346286 | A346259 |

k=8 | A346326 | A346327 | A346328 | A346329 | A346330 | A346331 | A346332 | A346333 | A346334 | A346335 |

k=9 | A346336 | A346337 | A346338 | A346339 | A346340 | A346341 | A346342 | A346343 | A346344 | A346345 |

k=10 | A346346 | A346347 | A346348 | A346349 | A346350 | A346351 | A346352 | A346353 | A346354 | A346355 |