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Index to Sums of like powers
From OeisWiki
Contents
- 1 Overview
- 2 Groups of sequences for like powers
- 3 New Lists
- 3.1 Squares
- 3.1.1 Numbers that can be expressed as the sum of k distinct squares in m or more ways (Table R2d)
- 3.1.2 Numbers that can be expressed as the sum of k possibly equal squares in m or more ways (Table R2e)
- 3.1.3 Numbers that can be expressed as the sum of k distinct squares in exactly m ways (Table S2d)
- 3.1.4 Numbers that can be expressed as the sum of k possibly equal squares in exactly m ways (Table S2e)
- 3.2 Cubes
- 3.2.1 Numbers that can be expressed as the sum of k distinct cubes in m or more ways (Table R3d)
- 3.2.2 Numbers that can be expressed as the sum of k possibly equal cubes in m or more ways (Table R3e)
- 3.2.3 Numbers that can be expressed as the sum of k distinct cubes in exactly m ways (Table S3d)
- 3.2.4 Numbers that can be expressed as the sum of k possibly equal cubes in exactly m ways (Table S3e)
- 3.3 4th powers
- 3.4 5th powers
- 3.5 Links
- 3.1 Squares
Overview
Numerous sequences in the OEIS are related to sums of the form a^m + b^m + c^m .... Such sums occur in a series of famous number-theoretical problems, among them Fermat's 4n+1 theorem, Euler's conjecture, Lagrange's four-square theorem, Waring's problem etc. (cf. the links section, below).
Groups of sequences for like powers
The sums can be grouped in different categories:
- an exact number k of summands is required
- there may be at most k summands
- the summands must all be different
The individual tables below list the OEIS sequences within these categories. When the cursor is moved over a sequence number, the name of the sequence is shown.
Sums of k m-th powers >= 0 (Table 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 (Table 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 (Table 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 (Table 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 (Table 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
Squares
Numbers that can be expressed as the sum of k distinct squares in m or more ways (Table R2d)
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 | 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 |
Numbers that can be expressed as the sum of k possibly equal squares in m or more ways (Table R2e)
m>=1 | m>=2 | m>=3 | m>=4 | m>=5 | m>=6 | m>=7 | m>=8 | m>=9 | m>=10 | |
---|---|---|---|---|---|---|---|---|---|---|
k=2 | A007692 | A025294 | A025295 | A025296 | A025297 | A025298 | A025299 | A025300 | A025301 | |
k=3 | A025331 | A025332 | A025333 | A025334 | A025335 | A025336 | A025337 | A025338 | ||
k=4 | A025367 | A025368 | A025369 | A025370 | A025371 | A025372 | A025373 | A025374 | A025375 | |
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 distinct squares in exactly m ways (Table S2d)
m=1 | m=2 | m=3 | m=4 | m=5 | m=6 | m=7 | m=8 | m=9 | m=10 | |
---|---|---|---|---|---|---|---|---|---|---|
k=2 | A025302 | A025303 | A025304 | A025305 | A025306 | A025307 | A025308 | A025309 | A025310 | A025311 |
k=3 | A025339 | A025340 | A025341 | A025342 | A025343 | A025344 | A025345 | A025346 | A025347 | A025348 |
k=4 | A025376 | A025377 | A025378 | A025379 | A025380 | A025381 | A025382 | A025383 | A025384 | A025385 |
Numbers that can be expressed as the sum of k possibly equal squares in exactly m ways (Table S2e)
m=1 | m=2 | m=3 | m=4 | m=5 | m=6 | m=7 | m=8 | m=9 | m=10 | m=11 | |
---|---|---|---|---|---|---|---|---|---|---|---|
k=2 | A025284 | A085625 | A025286 | A025287 | A294716 | A025289 | A025290 | A025291 | A025292 | A025293 | A236711 |
k=3 | A025321 | A025322 | A025323 | A025324 | A025325 | A025326 | A025327 | A025328 | A025329 | A025330 | |
k=4 | A025357 | A025358 | A025359 | A025360 | A025361 | A025362 | A025363 | A025364 | A025365 | A025366 | |
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 |
Cubes
Numbers that can be expressed as the sum of k distinct cubes in m or more ways (Table R3d)
m>=1 | m>=2 | m>=3 | |
---|---|---|---|
k=1 | A000578 | ||
k=2 | A001235 | ||
k=3 | A024974 | A025402 | |
k=4 | A259060 | A025413 |
Numbers that can be expressed as the sum of k possibly equal cubes in m or more ways (Table R3e)
m>=1 | m>=2 | m>=3 | m>=4 | m>=5 | m>=6 | m>=7 | m>=8 | m>=9 | m>=10 | |
---|---|---|---|---|---|---|---|---|---|---|
k=2 | A018787 | A023051 | A051167 | |||||||
k=3 | A025398 | A343968 | A343967 | A345083 | A345086 | A345087 | A345119 | A345121 | ||
k=4 | A003327 | A025406 | A025407 | 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 distinct cubes in exactly m ways (Table S3d)
m=1 | m=2 | m=3 | |
---|---|---|---|
k=3 | A025399 | A025400 | A025401 |
k=4 | A025408 | A025409 | A025410 |
Numbers that can be expressed as the sum of k possibly equal cubes in exactly m ways (Table S3e)
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 | A025395 | A025396 | A025397 | A343969 | A343970 | A345084 | A345085 | A345088 | A345120 | A345122 |
k=4 | A025403 | A025404 | A025405 | 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 |
4th powers
Numbers that can be expressed as the sum of k fourth powers in m or more ways (Table 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 (Table 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 |
5th powers
Numbers that can be expressed as the sum of k fifth powers in m or more ways (Table 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 (Table 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 |
Links
From MathWorld--A Wolfram Web Resource, by Weisstein, Eric W.:
- Euler's Sum of Powers Conjecture (with the extension to the Lander-Parkin-Selfridge conjecture)
- Lagrange's four-square theorem
- Taxicab Number
- Waring's problem
Other resources:
- Computing Minimal Equal Sums Of Like Powers, a collaborative website maintained by Jean-Charles Meyrignac