A003828 -id:A003828 - cp's OEIS frontend

A063923 Numbers k such that k^5 = a^5 + b^5 + c^5 + d^5 + e^5 has a nontrivial primitive solution in nonnegative integers.

72, 94, 107, 144, 365, 415, 427, 435, 480, 503, 530, 553, 575, 650, 700, 703, 716, 729, 744, 764, 804, 848, 851, 875, 923, 941, 975, 1004, 1006, 1040, 1044, 1235, 1257, 1313, 1327, 1329, 1369, 1392, 1457, 1469, 1504, 1528, 1537, 1575, 1583, 1588, 1596, 1623, 1653, 1685, 1686

Offset: 1

David W. Wilson, Aug 31 2001

nonn

Primitive means a solution for k has gcd(a,b,c,d,e) = 1. [Corrected by Jianing Song, Jan 24 2020]

Nontrivial means at least two of a,b,c,d,e are nonzero. - Jianing Song, Jan 24 2020

			   72^5 = 19^5 + 43^5 + 46^5 + 47^5 +  67^5;
   94^5 = 21^5 + 23^5 + 37^5 + 79^5 +  84^5;
  107^5 =  7^5 + 43^5 + 57^5 + 80^5 + 100^5.

Jianing Song, Table of n, a(n) for n = 1..400 (All terms from James Waldby link)
L. J. Lander, T. R. Parkin and J. L. Selfridge, A Survey of Equal Sums of Like Powers, Mathematics of Computation, Vol. 21, No. 99 (Jul., 1967), pp. 446-459. See Table III p. 449.
James Waldby, A Table of Fifth Powers equal to a Fifth Power
Index to sequences related to Diophantine equations (5,1,5)

Cf. A063922.
For cubes: A003072, A023041, A261029.
For fourth powers: A003828, A175610, A039664, A003294.

144 and 1006 inserted and name simplified by Jianing Song, Jan 24 2020

More terms from Jinyuan Wang, Jan 24 2020

A175610 Numbers k such that k^4 = x^4 + y^4 + z^4, where x,y,z are positive integers.

Original entry on oeis.org

422481, 844962, 1267443, 1689924, 2112405, 2534886, 2813001, 2957367, 3379848, 3802329, 4224810, 4647291, 5069772, 5492253, 5626002, 5914734, 6337215, 6759696, 7182177, 7604658, 8027139, 8439003, 8449620, 8707481, 8872101, 9294582, 9717063, 10139544, 10562025, 10984506

Offset: 1

Juri-Stepan Gerasimov, Jul 24 2010

nonn

Main sequence is A003828.

			a(1) = 422481 because 422481^4 = 95800^4 + 217519^4 + 414560^4.
a(61) = 20615673 because 20615673^4 = 2682440^4 + 15365639^4 + 18796760^4.

Charles R Greathouse IV, Table of n, a(n) for n = 1..4999

Cf. A003337, A003828, A175598.

Select[Range[100], (p = PowersRepresentations[#^4, 3, 4]; (Select[p, #[[1]] > 0 && #[[2]] > 0 && #[[3]] > 0 &] != {})) &] (* Jinyuan Wang, Feb 20 2020 *)

is(n) = for(a=sqrtnint(n^4\3,4), n-1, for(b=1, a, for(c=1, b, if(n^4==a^4+b^4+c^4, return(1))))); 0; \\ Charles R Greathouse IV, Aug 29 2013 and slightly modified by Jinyuan Wang, Feb 20 2020

Terms and example corrected by Charles R Greathouse IV, Aug 29 2013

First 0 removed by Jinyuan Wang, Feb 20 2020

A063922 Numbers k such that k^5 = a^5 + b^5 + c^5 + d^5 + e^5 has a nontrivial solution in nonnegative integers.

Original entry on oeis.org

72, 94, 107, 144, 188, 214, 216, 282, 288, 321, 360, 365, 376, 415, 427, 428, 432, 435, 470, 480, 503, 504, 530, 535, 553, 564, 575, 576, 642, 648, 650, 658, 700, 703, 716, 720, 729, 730, 744, 749, 752, 764, 792, 804, 830, 846, 848, 851, 854, 856, 864, 870

Offset: 1

David W. Wilson, Aug 31 2001

nonn

Any multiple of a term is again a term of this sequence. See A063923 for the primitive solutions. See A007666 for similar solutions for other powers. - M. F. Hasler, Nov 17 2015

Nontrivial means at least two of a,b,c,d,e are nonzero. - Jianing Song, Jan 24 2020

			   72^5 = 19^5 + 43^5 + 46^5 + 47^5 +  67^5;
   94^5 = 21^5 + 23^5 + 37^5 + 79^5 +  84^5;
  107^5 =  7^5 + 43^5 + 57^5 + 80^5 + 100^5.

Cf. A063923.

For fourth powers: A003828, A175610, A039664, A003294.

A134341 Numbers whose fifth powers have a partition as a sum of fifth powers of four positive integers.

Original entry on oeis.org

144, 288, 432, 576, 720, 864, 1008, 1152, 1296, 1440, 1584, 1728, 1872, 2016, 2160, 2304, 2448, 2592, 2736, 2880, 3024, 3168, 3312, 3456, 3600, 3744, 3888, 4032, 4176, 4320, 4464, 4608, 4752, 4896, 5040, 5184, 5328, 5472, 5616, 5760, 5904, 6048, 6192

Offset: 1

Artur Jasinski, Oct 21 2007

nonn

The only primitive terms (that is, in which the summands do not all have a common factor) known are 144 and 85359. - Jianing Song, Jan 24 2020

The paper by Lander and Parkin where they just give the first known counterexample to Euler's conjecture, 27^5 + 84^5 + 110^5 + 133^5 = 144^5, found using a CDC6600, is known as one of the shortest published proofs. - M. F. Hasler, Mar 11 2020

			a(1) = 144 because 144^5 = 27^5 + 84^5 + 110^5 + 133^5;
a(593) = 85359 because 85359^5 = 55^5 + 3183^5 + 28969^5 + 85282^5 = 4531548087264753520490799 (Jim Frye 2005). [Typo corrected by _Sébastien Palcoux_, Jul 05 2017]

L. E. Dickson, History of the theory of numbers, Vol. 2, Chelsea, New York, 1952, p. 648.

L. J. Lander and T. R. Parkin, Counterexample to Euler's conjecture on sums of like powers, Bull. Amer. Math. Soc. 72 (6) (1966), p. 1079.
Burkard Polster, Euler's and Fermat's last theorems, the Simpsons and CDC6600, Mathologer video (2018).
Wikipedia, Euler's sum of powers conjecture
Index to sequences related to Diophantine equations (5,1,4)

Cf. A063923, A063922, A003828, A175610.

Incorrect formula removed by Jianing Song, Jan 24 2020

A078518 Consider primitive solutions (x,y,z) to x^4+y^4+z^4=w^4 with 0

Original entry on oeis.org

95800, 2767624, 8332208, 11289040, 12552200, 3642840, 2682440, 2164632, 10409096, 34918520, 1841160, 27450160, 186668000, 219076465, 558424440, 588903336, 50237800, 686398000, 92622401

Offset: 1

N. J. A. Sloane, Jan 07 2003

See A003828 for references and links.

Cf. A003828, A078519, A078520.

a(8)-a(19) from A003828 added by Sean A. Irvine, Jul 03 2025

A078519 Consider primitive solutions (x,y,z) to x^4+y^4+z^4=w^4 with 0

Original entry on oeis.org

217519, 1390400, 5507880, 8282543, 12552200, 7028600, 15365639, 31669120, 42878560, 87865617, 121952168, 108644015, 260052385, 275156240, 606710871, 859396455, 632671960, 1237796960, 1553556440

Offset: 1

N. J. A. Sloane, Jan 07 2003

See A003828 for references and links.

Cf. A003828, A078518, A078520.

Data corrected and extended using A003828 by Sean A. Irvine, Jul 03 2025

A078520 Consider primitive solutions (x,y,z) to x^4+y^4+z^4=w^4 with 0

Original entry on oeis.org

414560, 2767624, 8332208, 11289040, 14173720, 16281009, 18796760, 41084175, 65932985, 106161120, 122055375, 146627384, 582665296, 630662624, 769321280, 1166705840, 1670617271, 1662997663, 1593513080

Offset: 1

N. J. A. Sloane, Jan 07 2003

See A003828 for references and links.

Cf. A003828, A078518, A078519.

Data corrected and extended using A003828 by Sean A. Irvine, Jul 03 2025

A130022 Smallest natural number whose 4th power is the sum of n 4th powers of distinct natural numbers, or 0 if no such number exists.

Original entry on oeis.org

1, 0, 422481, 353, 15, 35, 25, 31, 37, 41, 35, 43, 39, 43, 47, 53, 55, 50, 50, 46, 48, 48, 50, 48, 50, 48, 52, 53, 55, 56, 54, 58, 58, 63, 65, 67, 70, 71, 73, 77, 81, 85, 87, 91, 93, 97, 101

Offset: 1

J. Lowell, Jun 15 2007

Cf. a(3) A003828, a(4) A096739, 3rd powers A130012, n-th powers A007666.

More terms from Martin Fuller, Jul 06 2007

A331675 Numbers k such that k^4 = a^4 + b^4 + c^4 + d^4 has at least two positive primitive solutions.

Original entry on oeis.org

31127, 41963, 72899, 154789, 195479, 208471

Offset: 1

Jianing Song, Jan 24 2020

Primitive solutions means gcd(a,b,c,d) = 1.

These are all terms from Jaroslaw Wroblewski link, which gives all positive solutions to k^4 = a^4 + b^4 + c^4 + d^4 where k < 222000, gcd(a,b,c,d) = 1.

			Solutions to k^4 = a^4 + b^4 + c^4 + d^4 = a'^4 + b'^4 + c'^4 + d'^4:
31127: (2260, 4870, 17386, 30335), (2495, 11998, 16430, 30320);
41963: (1100, 17260, 25015, 40234), (8750, 12109, 14470, 41720);
72899: (4555, 44270, 58868, 59330), (9700, 16480, 47618, 69265);
154789: (49586, 55450, 102170, 145615), (66405, 106740, 119760, 121664);
195479: (12970, 43340, 140947, 180520), (25570, 41080, 112822, 189695);
208471: (3903, 46560, 61290, 207950), (91045, 149222, 150550, 168730).

Jaroslaw Wroblewski, Exhaustive list of 1009 solutions to (4,1,4) below 222,000 (Note: (t,m,n) denotes the equation Sum_{i=1..m} (a_i)^t = Sum_{j=1..n} (b_j)^t, where a_i, b_j are positive integers, gcd(a_1, a_2, ..., a_m, b_1, b_2, ..., b_n) = 1.)

Subsequence of A039664 (and thus of A003294).
Other similar sequences:
A023041 (k^3=a^3+b^3+c^3, gcd(a,b,c)=1);
A003828 (k^4=a^4+b^4+c^4, gcd(a,b,c)=1);
A175610 (k^4=a^4+b^4+c^4);
A134341 (k^5=a^5+b^5+c^5+d^5);
A063923 (k^5=a^5+b^5+c^5+d^5+e^5, gcd(a,b,c,d,e)=1);
A063922 (k^5=a^5+b^5+c^5+d^5+e^5);
A331674 (k^5=a^5+b^5+c^5+d^5+e^5, gcd(a,b,c,d,e)=1, at least two solutions).

A347773 Square array read by antidiagonals downwards: T(n,k) is the smallest positive integer whose n-th power is the sum of k n-th powers of positive integers, or 0 if no such number exists.

Original entry on oeis.org

1, 2, 1, 3, 5, 1, 4, 3, 0, 1, 5, 2, 6, 0, 1, 6, 4, 7, 422481, 0, 1, 7, 3, 4, 353

Offset: 1

Eric Chen, Sep 15 2021

a(26) = T(5,3) is conjectured to be 0, but this has not been proved.
By Fermat's last theorem, T(n,2) = 0 for n > 2.
Euler's sum of powers conjecture is that T(n,k) = 0 for n > k > 1, but this conjecture is not true: T(4,3) = 422481, T(5,4) = 144, there are no known counterexamples for n >= 6.
There are no known 0s for k > 2.
Conjecture: If T(n,k) = 0, then T(r,k) = T(n,s) = T(r,s) = 0 for all r >= n, 2 <= s <= k.

			Table begins:
  n\k |  1   2       3    4   5   6     7     8
  ----+----------------------------------------
   1  |  1   2       3    4   5   6     7     8
   2  |  1   5       3    2   4   3     4     4
   3  |  1   0       6    7   4   3     5     2
   4  |  1   0  422481  353   5   3     9    13
   5  |  1   0       ?  144  72  12    23    14
   6  |  1   0       ?    ?   ?   ?  1141   251
   7  |  1   0       ?    ?   ?   ?   568   102
   8  |  1   0       ?    ?   ?   ?     ?  1409
T(2,5) = 4 because 4^2 = 1^2 + 1^2 + 1^2 + 2^2 + 3^2 and there is no smaller square that is the sum of 5 positive squares.
T(4,3) = 422481 because 422481^4 = 95800^4 + 217519^4 + 414560^4 and there is no smaller 4th power that is the sum of 3 positive 4th powers.
T(7,7) = 568 because 568^7 = 127^7 + 258^7 + 266^7 + 413^7 + 430^7 + 439^7 + 525^7 and there is no smaller 7th power that is the sum of 7 positive 7th powers.

Ed Pegg Jr., Power Sums, Math Games, November 13 2006.
James Waldby, A Table of Fifth Powers equal to Sums of Five Fifth Powers, 2009.
Eric Weisstein's World of Mathematics, Euler's sum of powers conjecture
Eric Weisstein's World of Mathematics, Diophantine Equation--3rd Powers
Eric Weisstein's World of Mathematics, Diophantine Equation--4th Powers
Eric Weisstein's World of Mathematics, Diophantine Equation--5th Powers
Eric Weisstein's World of Mathematics, Diophantine Equation--6th Powers
Eric Weisstein's World of Mathematics, Diophantine Equation--7th Powers
Eric Weisstein's World of Mathematics, Diophantine Equation--8th Powers
Wikipedia, Euler's sum of powers conjecture

Cf. A007666 (main diagonal), A264764 (subdiagonal for k = n-1).
Cf. A175610 and A003828 (both for a(19) = T(4,3) = 422481).
Cf. A003294 and A039664 (both for a(25) = T(4,4) = 353).
Cf. A134341 (for a(33) = T(5,4) = 144).
Cf. A063922 and A063923 (both for a(41) = T(5,5) = 72).
Cf. A130012, A130022 (these two sequences are not rows of this table, since they require DISTINCT n-th powers, but this table does not have that requirement).

/* return 0 instead of 1 for n=1, and oo loop when T(n, k)=0 */ A347773(p, n, s, m)={ /* Check whether s can be written as sum of n positive p-th powers not larger than m^p. If so, return the base a of the largest term a^p. */ s>n*m^p && return; n==1&&return(ispower(s, p, &n)*n); /* if s and m are not given, s>=n and m are arbitrary. */ !s&&for(m=round(sqrtn(n, p)), 9e9, A347773(p, n, m^p, m-1)&&return(m)); for(a=ceil(sqrtn(s\n, p)), min(sqrtn(max(0, s-n+1), p), m), A347773(p, n-1, s-a^p, a)&&return(a)); } /* after M. F. Hasler in A007666 */ /* Just enter "A347773(n, k)" to get T(n, k) */

T(n,1) = 1.
T(1,k) = k.
T(n,2) = 0 for n > 2.
T(n,n) = A007666(n).
T(n,n-1) = A264764(n).
T(3,k) <= A130012(k).
T(4,k) <= A130022(k).

cp's OEIS Frontend

A063923 Numbers k such that k^5 = a^5 + b^5 + c^5 + d^5 + e^5 has a nontrivial primitive solution in nonnegative integers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A175610 Numbers k such that k^4 = x^4 + y^4 + z^4, where x,y,z are positive integers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A063922 Numbers k such that k^5 = a^5 + b^5 + c^5 + d^5 + e^5 has a nontrivial solution in nonnegative integers.

Views

Author

Keywords

Comments

Examples

Crossrefs

A134341 Numbers whose fifth powers have a partition as a sum of fifth powers of four positive integers.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Extensions

A078518 Consider primitive solutions (x,y,z) to x^4+y^4+z^4=w^4 with 0

Views

Author

Keywords

References

Crossrefs

Extensions

A078519 Consider primitive solutions (x,y,z) to x^4+y^4+z^4=w^4 with 0

Views

Author

Keywords

References

Crossrefs

Extensions

A078520 Consider primitive solutions (x,y,z) to x^4+y^4+z^4=w^4 with 0

Views

Author

Keywords

References

Crossrefs

Extensions

A130022 Smallest natural number whose 4th power is the sum of n 4th powers of distinct natural numbers, or 0 if no such number exists.

Views

Author

Keywords

Crossrefs

Extensions

A331675 Numbers k such that k^4 = a^4 + b^4 + c^4 + d^4 has at least two positive primitive solutions.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A347773 Square array read by antidiagonals downwards: T(n,k) is the smallest positive integer whose n-th power is the sum of k n-th powers of positive integers, or 0 if no such number exists.

Views

Author

Keywords

Comments

Examples