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imported>Gfis Created page with "==Various Topics== * I recently got an [https://www.rechnen-ohne-strom.de/rechner-galerie/zahlenschieber/addimult/ ADDIMULT]." |
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== | === Sums of like powers=== | ||
==== Sums of k m-th powers >= 0==== | |||
{| 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 self-avoiding n-step walks on 4-d cubic lattice with first step specified, A242355(n)/8.">[https://oeis.org/A323856 A323856]</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 4th powers of the degrees of irreducible representations of S_n, the symmetric group on n letters.">[https://oeis.org/A129627 A129627]</span>||<span title="Sum of 5th powers: 0^5 + 1^5 + 2^5 + ... + n^5.">[https://oeis.org/A000539 A000539]</span>||<span title="Sum of 6th powers: 0^6 + 1^6 + 2^6 + ... + n^6.">[https://oeis.org/A000540 A000540]</span>||<span title="Sum of 7th powers: 1^7 + 2^7 + ... + n^7.">[https://oeis.org/A000541 A000541]</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 10th powers.">[https://oeis.org/A023002 A023002]</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==== | |||
{| 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 positive 5th powers.">[https://oeis.org/A123294 A123294]</span>|| || || || || ||  | |||
|- | |||
| k=14 || || || ||<span title="Sum of 14 positive 5th powers.">[https://oeis.org/A123295 A123295]</span>|| || || || || ||  | |||
|- | |||
|} | |||
==== Sums of at most k positive m-th powers > 0==== | |||
{| 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==== | |||
{| 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 n having exactly k representations as the sum of m squares >= 0==== | |||
<!--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> | |||
|- | |||
|} | |||
Revision as of 15:08, 7 August 2020
Sums of like powers
Sums of k m-th powers >= 0
| 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 | A323856 | A294287 | A129627 | A000539 | A000540 | A000541 | A155468 | A007487 | A023002 | A181134 | ||
| k=2 | A140328 | A004999 | A018786 | |||||||||
| k=3 | A294713 | A332201 | A193244 | |||||||||
| k=4 | A001245 |
Sums of exactly k positive m-th powers > 0
| 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 | A123294 | |||||||||
| k=14 | A123295 |
Sums of at most k positive m-th powers > 0
| 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
| m=2 | m=3 | |
|---|---|---|
| k=-1 | A078131 | |
| k=2 | A294073 | |
| k=3 | A302359 | A302360 |
Numbers n having exactly k representations as the sum of m squares >= 0
| 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 |