A063922 -id:A063922 - 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

A007666 a(n) = smallest number k such that k^n is the sum of n positive n-th powers, or 0 if no solution exists.

Original entry on oeis.org

1, 5, 6, 353, 72

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

The next term a(6) has been claimed to be 1141, but this is incorrect. In fact, 1141^6 is the sum of seven 6th powers. - Jud McCranie, Jun 10 2007
a(7) = 568 and a(8) = 1409. - J. Lowell, Jul 25 2007
a(6) is either 0 (no solution) or greater than 730000 (see the Resta & Meyrignac link, p. 1054). - Jon E. Schoenfield, Jul 22 2017

			1^1 = 1^1.
5^2 = 3^2 + 4^2.
6^3 = 3^3 + 4^3 + 5^3.
353^4 = 30^4 + 120^4 + 272^4 + 315^4.
72^5 = 19^5 + 43^5 + 46^5 + 47^5 + 67^5.
568^7 = 127^7 + 258^7 + 266^7 + 413^7 + 430^7 + 439^7 + 525^7.
1409^8 = 90^8 + 223^8 + 478^8 + 524^8 + 748^8 + 1088^8 + 1190^8 + 1324^8.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 164.

Jean-Charles Meyrignac, FAQ File
Jean-Charles Meyrignac, Resta's Algorithm
Giovanni Resta and Jean-Charles Meyrignac, The smallest solutions to the diophantine equation x^6 + y^6 = a^6 + b^6 + c^6 + d^6 + e^6, Math. Comp. 72 (2003), pp. 1051-1054.

k^n = T(n, 1)^n + ... + T(n, n)^n, where T() is given in A061988.
Examples for n=4 are in A003294.
Examples for n=5 are in A063922.

A007666(n,s,m,p=n)={ /* 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,m,p are not given, s>=n and m are arbitrary and p=n. */ !s&&for(m=round(sqrtn(n,n)),9e9,A007666(n,m^n,m-1,n)&&return(m)); for(a=ceil(sqrtn(s\n,p)),min(sqrtn(s-n+1,p),m),A007666(n-1,s-a^p,a,p)&&return(a));} \\ M. F. Hasler, Nov 17 2015

Name clarified by Dmitry Kamenetsky, Aug 05 2015

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

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).

A134297 a(n) = 107*n.

Original entry on oeis.org

0, 107, 214, 321, 428, 535, 642, 749, 856, 963, 1070, 1177, 1284, 1391, 1498, 1605, 1712, 1819, 1926, 2033, 2140, 2247, 2354, 2461, 2568, 2675, 2782, 2889, 2996, 3103, 3210, 3317, 3424, 3531, 3638, 3745, 3852, 3959, 4066, 4173, 4280, 4387, 4494, 4601, 4708

Offset: 0

Artur Jasinski, Oct 18 2007

nonn

For n > 0, a(n)^5 has a partition as the sum of fifth powers of five positive numbers: (107n)^5 = (7n)^5 + (43n)^5 + (57n)^5 + (80n)^5 + (100n)^5. [Corrected by Jianing Song, Jan 24 2020]

			107^5 = 7^5 + 43^5 + 57^5 + 80^5 + 100^5.

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A134298, A063923.

Without the initial 0, subsequence of A063922 (k such that k^5 = a^5+b^5+c^5+d^5+e^5, where at least two of a,b,c,d,e are nonzero).

Mathematica
```
Table[107n, {n, 0, 30}]
```

a(n) = 107*n \\ Jianing Song, Jan 24 2020

G.f.: 107*x/(-1+x)^2. - R. J. Mathar, Nov 14 2007

a(0) prepended by Jianing Song, Jan 24 2020

A331674 Numbers k such that k^5 = a^5 + b^5 + c^5 + d^5 + e^5 has at least two primitive solutions in nonnegative integers.

Original entry on oeis.org

744, 1686, 1921, 2087, 3447, 4097, 6065, 7157, 7864, 8570

Offset: 1

Jianing Song, Jan 24 2020

Primitive solutions means gcd(a,b,c,d,e) = 1.
These are all terms from James Waldby link, which gives all solutions to k^5 = a^5 + b^5 + c^5 + d^5 + e^5 where k < 10000, gcd(a,b,c,d,e) = 1 and at least two of a,b,c,d,e are nonzero.
Note that if nonprimitive solutions were allowed (where at least two of a,b,c,d,e are nonzero), then 144 would be a term because 144^5 = 0^5 + 27^5 + 84^5 + 110^5 + 133^5 = 38^5 + 86^5 + 92^5 + 94^5 + 134^5.

			Solutions to k^5 = a^5 + b^5 + c^5 + d^5 + e^5 = a'^5 + b'^5 + c'^5 + d'^5 + e'^5:
744: (100, 210, 414, 629, 651), (14, 95, 545, 586, 644);
1686: (265, 486, 784, 791, 1670), (46, 591, 675, 999, 1655);
1921: (275, 351, 872, 1298, 1855), (95, 771, 1020, 1519, 1756);
2087: (145, 565, 1105, 1462, 1990), (519, 642, 1026, 1480, 1990);
3447: (1212, 1300, 1345, 1699, 3411), (289, 317, 1033, 1682, 3426);
4097: (1281, 2154, 2396, 3462, 3504), (954, 1989, 2127, 2396, 3981);
6065: (3629, 3811, 4070, 4272, 5313), (854, 3160, 3752, 5073, 5196);
7157: (1827, 2186, 4789, 5629, 6376), (930, 2746, 3570, 5109, 6802);
7864: (1093, 2309, 3629, 6137, 7296), (312, 1631, 3418, 3544, 7809);
8570: (1766, 2529, 4086, 5520, 8319), (2101, 2315, 2710, 3960, 8524).

James Waldby, A Table of Fifth Powers equal to a Fifth Power

Subsequence of A063923 (and thus of A063922).
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);
A039664 (k^4=a^4+b^4+c^4+d^4, gcd(a,b,c,d)=1);
A003294 (k^4=a^4+b^4+c^4+d^4);
A331675 (k^4=a^4+b^4+c^4+d^4, gcd(a,b,c,d)=1, at least two solutions).
A134341 (k^5=a^5+b^5+c^5+d^5).

A386672 Integers x such that there exist four integers 0

Original entry on oeis.org

46, 94, 946, 1139, 1680, 3804, 4200, 29975, 31143, 48560, 53428, 63840, 74178, 121400, 125280, 135720, 279300, 483392, 679952

Offset: 1

S. I. Dimitrov, Jul 28 2025

The numbers x, y, z, t and w form a sigma-quintic quintuple.

Other terms of the sequence: 5446350, 20201728, 326481408.

			(46, 19, 43, 47, 67) is such a quintuple because sigma(46)^5 = 72^5 = 46^5 + 19^5 + 43^5 + 47^5 + 67^5.
Other examples: (94, 38, 86, 92, 134), (946, 418, 1012, 1034, 1474), (1139, 323, 731, 782, 799), (63840, 144480, 154560, 157920, 225120).

S. I. Dimitrov, Generalizations of amicable numbers, arXiv:2408.07387 [math.NT], 2024.
James Waldby, A Table of Fifth Powers equal to a Fifth Power

Cf. A000203, A003350, A063922, A386225.

More terms from Michel Marcus, Jul 29 2025

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.

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A007666 a(n) = smallest number k such that k^n is the sum of n positive n-th powers, or 0 if no solution exists.

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

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

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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.

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A134297 a(n) = 107*n.

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A331674 Numbers k such that k^5 = a^5 + b^5 + c^5 + d^5 + e^5 has at least two primitive solutions in nonnegative integers.

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A386672 Integers x such that there exist four integers 0

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