Index to Sums of like powers

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