A063923 -id:A063923 - cp's OEIS frontend

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

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

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).

A376289 Values k for primitive solutions to k^5 + a^5 + b^5 = c^5 + d^5 + e^5 with k >= a >= b >= 0 and k > c >= d >= e >= 0, repetitions allowed.

Original entry on oeis.org

66, 67, 74, 83, 107, 118, 119, 123, 136, 142, 152, 155, 169, 170, 181, 182, 186, 201, 204, 215, 216, 224, 229, 233, 234, 248, 258, 264, 274, 282, 283, 286, 288, 289, 293, 294, 307, 310, 310, 328, 331, 348, 364, 364, 373, 377, 378, 394, 399, 413, 417, 420, 421, 425, 426, 430, 430, 433, 436, 448, 459, 470, 474, 480, 486, 490, 498

Offset: 1

Artur Jasinski, Sep 19 2024

nonn

This case is known in literature as 5.3.3 (see e.g. Eric Weisstein's World of Mathematics).
Primitive means a solution has gcd(k,a,b,c,d,e) = 1.
For primitive solutions of the 5.1.5 case see A063923.
For primitive solutions of the 5.2.4 case see A376914.
Although the definition does not require all coefficients to be nonzero or distinct, all known solutions have k > a > b > 0 and c > d > e > 0.
In every known case, k+a+b-c-d-e is even and very often zero.
This sequence is infinite as follows:
1) Bremner's modified one parameter identity (with conditions k+a+b-c-d-e=0 and k-a=c-d):
  (37888 + 67978*w + 53683*w^2 + 24217*w^3 + 6750*w^4 + 1164*w^5 + 115*w^6 + 5*w^7)^5+
  (15744 + 33046*w + 29861*w^2 + 15193*w^3 + 4738*w^4 + 912*w^5 + 101*w^6 + 5*w^7)^5+
  (16376 + 33534*w + 29739*w^2 + 14937*w^3 + 4622*w^4 + 888*w^5 + 99*w^6 + 5*w^7)^5
  =
  (27912 + 52390*w + 43165*w^2 + 20281*w^3 + 5882*w^4 + 1056*w^5 + 109*w^6 + 5*w^7)^5+
  (5768 + 17458*w + 19343*w^2 + 11257*w^3 + 3870*w^4 + 804*w^5 + 95*w^6 + 5*w^7)^5+
  (36328 + 64710*w + 50775*w^2 + 22809*w^3 + 6358*w^4 + 1104*w^5 + 111*w^6 + 5*w^7)^5
  which generate members of this sequence for nonnegative w=0,1,2,3,...
2) Moessner's one parameter identity (k+a+b-c-d-e=40*n)
  (a^36 + 8*a^26 + 12*a^16 + 20*a^11 - a^6)^5+
  (a^33 - 12*a^23 - 28*a^13 - a^3)^5+
  (a^30 + 20*a^25 - 12*a^20 - 8*a^10 - 1)^5
  =
  (a^36 + 8*a^26 + 12*a^16 - 20*a^11 - a^6)^5+
  (a^33 + 28*a^23 + 12*a^13 - a^3)^5+
  (a^30 - 20*a^25 - 12*a^20 - 8*a^10 - 1)^5
  which generate members of this sequence for a=2,3,4,...
3) Moessner's two parameter identity (with condition k+a+b-c-d-e=0):
  (75*x^7-230*x^6*y-113*x^5*y^2+510*x^4*y^3-407*x^3*y^4+62*x^2*y^5+125*x*y^6-150*y^7)^5+
  (-175*x^7+170*x^6*y-391*x^5*y^2-30*x^4*y^3+451*x^3*y^4-602*x^2*y^5+115*x*y^6-50*y^7)^5+
  (175*x^7-160*x^6*y+387*x^5*y^2-108*x^4*y^3+5*x^3*y^4-336*x^2*y^5+265*x*y^6-100*y^7)^6
  =
  (25*x^7-290*x^6*y+689*x^5*y^2-138*x^4*y^3+27*x^3*y^4-62*x^2*y^5+155*x*y^6-150*y^7)^5+
  (-25*x^7-653*x^5*y^2+564*x^4*y^3-195*x^3*y^4-208*x^2*y^5+105*x*y^6-100*y^7)^5+
  (75*x^7+70*x^6*y-153*x^5*y^2-54*x^4*y^3+217*x^3*y^4-606*x^2*y^5+245*x*y^6-50*y^7)^5
4) Choudhry and Wróblewski two parameter identity:
  (2 p^15 q + 6 p^5 q^11)^5 +
  (p^16 - 3 p^11 q^5 - 5 p^6 q^10 - p q^15)^5 +
  (6 p^11 q^5 + 2 p q^15)^5
  = (p^16 + 3 p^11 q^5 - 5 p^6 q^10 + p q^15)^5 +
  (p^15 q + 5 p^10 q^6 + 3 p^5 q^11 - q^16)^5 +
  (p^15 q - 5 p^10 q^6 + 3 p^5 q^11 + q^16)^5
5) Edward Brisse two parameter identity (with condition k+a+b-c-d-e=0):
  (2*a^8*b+10*a^7*b^2-20*a^6*b^3+20*a^5*b^4-34*a^4*b^5-10*a^3*b^6+270*a^2*b^7-20*a*b^8+682*b^9)^5+
  (-2*a^8*b+10*a^7*b^2+20*a^6*b^3+20*a^5*b^4+34*a^4*b^5-10*a^3*b^6-270*a^2*b^7-20*a*b^8-682*b^9)^5+
  (a^9-22*a^5*b^4-125*a^3*b^6-79*a*b^8)^5
   =
  (a^8*b+10*a^7*b^2-10*a^6*b^3+20*a^5*b^4-92*a^4*b^5-160*a^3*b^6-15*a^2*b^7-320*a*b^8+341*b^9)^5+
  (-a^8*b+10*a^7*b^2+10*a^6*b^3+20*a^5*b^4+92*a^4*b^5-160*a^3*b^6+15*a^2*b^7-320*a*b^8-341*b^9)^5+
  (a^9-22*a^5*b^4+175*a^3*b^6+521*a*b^8)^5
When we take b=1 in this identity we obtain the Lander 1968 one parameter identity.

			67^5 + 28^5 + 24^5 = 62^5 + 54^5 + 3^5 so 67 is a term.
399^5 + 237^5 + 62^5 = 382^5 + 307^5 + 9^5 so 399 is a term.
310^5 + 118^5 + 102^5 = 271^5 + 270^5 + 49^5 and 310^5 + 124^5 + 116^5 = 294^5 + 235^5 + 21^5 so 310 is a repeated term.

A. Bremner, A geometric approach to equal sums of fifth powers, Number Th. 13, 337-354, 1981.
Edward Brisse in Jean-Charles Meyrignac Identities Of Equal Sums Of Like Power, Computing Minimal Equal Sums Of Like Powers 2001.
A. Choudhry and J. Wróblewski, A quintic diophantine equation with applications to two diophantine systems concerning fifth powers, Rocky Mountain J. Math. 43(6): 1893-1899 (2013).
Andrew Howroyd, Solutions for k <= 500, Oct 2024.
L. J. Lander, Geometric aspects of diophantine equations involving equal sums of like powers, American Mathematical Monthly Volume 75 no 6-10 1968 pp. 1061-1073
L. J. Lander, T. R. Parkin, and J. L. Selfridge, A Survey of Equal Sums of Like Powers, Math. Comput. 21, 446-459, 1967 (Table VI).
A. Moessner, Due sistemi diofantei., Bollettino dell'Unione Matematica Italiana Serie 3 6 (1951), p. 117-118 (Italian) Sezione Scientifica
Eric Weisstein's World of Mathematics, Diophantine Equation-5th Powers.
Wikipedia, Euler's sum of powers conjecture.

Cf. A063923, A345010, A376914.

ww = {}; Monitor[Do[Do[Do[Do[kk = PowersRepresentations[e^5 + d^5 + c^5 - k^5, 2, 5];
If[kk != {},If[GCD[k, c, d, e, kk[[1]][[1]], kk[[1]][[2]]] == 1,
AppendTo[ww, k]; Print[k];Print[{k, kk[[1]][[2]], kk[[1]][[1]], c, d, e}]]], {e, 0, d}],{d, 0, c}], {c, 0, k - 1}], {k, 4, 186}], {c, k}];ww

lista(maxk, prfull=0)={for(k=1, maxk, for(a=0, k, for(b=0, a, my(s=k^5+a^5+b^5); for(c=sqrtnint(s\3,5), k-1, for(d=sqrtnint((s-c^5-1)\2,5)+1, min(c, sqrtnint(s-c^5,5)), my(e); if(ispower(s-c^5-d^5,5,&e) && gcd([k,a,b,c,d,e])==1, if(prfull, print([k,a,b,c,d,e]), print1(k, ", ") )) )))))} \\ Andrew Howroyd, Oct 08 2024

A376914 Values k for primitive solutions to k^5 + a^5 + b^5 + c^5 = d^5 + e^5 with k >= a >= b >= c > 0 and d >= e >= 0, repetitions allowed.

Original entry on oeis.org

28, 37, 50, 63, 82, 86, 94, 99, 100, 102, 104, 112, 114, 129, 130, 133, 135, 137, 156, 172, 174, 184, 191, 196, 200, 213, 221, 236, 237, 241, 252, 258, 260, 270, 271, 279, 282, 291, 291, 291

Offset: 1

Artur Jasinski, Oct 09 2024

Primitive means a solution has gcd(k,a,b,c,d,e) = 1.
In most of cases d > k.
This case is known in literature as 5.2.4 (see e.g. Eric Weisstein's World of Mathematics).

			28^5 + 20^5 + 10^5 + 4^5 = 29^5 + 3^5 so 28 is a term.
133^5 + 110^5 + 84^5 + 27^5 = 144^5 + 0^5 so 133 is a term.
291^5 + 109^5 + 31^5 + 29^5 = 287^5 + 173^5 and 291^5 + 279^5 + 108^5 + 85^5 = 328^5 + 15^5 and 291^5 + 287^5 + 205^5 + 174^5 = 335^5 + 202^5 so 291 is included three times.

Andrew Howroyd, Solutions for k <= 300, Oct 2024.
Eric Weisstein's World of Mathematics, Diophantine Equation-5th Powers.
Wikipedia, Euler's sum of powers conjecture.

Cf. A063923, A134341, A376289.

aa = {}; Monitor[Do[Do[Do[Do[kk = PowersRepresentations[k^5 + a^5 + b^5 + c^5, 2, 5];If[kk != {}, If[GCD[k,a,b,c,kk[[1]][[1]],kk[[1]][[2]]]==1,Print[{k, a, b, c, kk}]; AppendTo[aa, k]]], {c, 1, b}], {b, 1, a}], {a, 1, k}], {k, 1, 200}], {a, k}]; aa

lista(maxk, mink=1,prfull=0)={for(k=mink, maxk, for(a=1, k, for(b=1, a, for(c=1,b,my(s=k^5+a^5+b^5+c^5);for(d=sqrtnint((s-1)\2,5)+1,  sqrtnint(s,5), my(e); if(ispower(s-d^5,5,&e) && gcd([k,a,b,c,d,e])==1, if(prfull, print([k,a,b,c,d,e]), print1(k, ", ") )) )))))} \\ Andrew Howroyd, Oct 09 2024

a(26)-a(40) from Andrew Howroyd, Oct 09 2024

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).

A350430 a(n) is the smallest n-th power which can be represented as the sum of n distinct positive n-th powers in exactly n ways, or -1 if none exists.

Original entry on oeis.org

1, 625, 157464

Offset: 1

Ilya Gutkovskiy, Dec 30 2021

From Jon E. Schoenfield, Dec 30 2021: (Start)
222000^4 < a(4) < 4891341^4 = lcm(2829, 12259, 16359, 30381)^4 (see A039664, including the Wroblewski link).
10000^5 <= a(5) < 12528^5 = lcm(72, 1044, 1392, 2088, 3132)^5 (see A063923, including the Waldby link; note that, although the terms of A063923 include 72, 144, 1044, 1392, and 2088, whose LCM is only 4176, the primitive solution in which the sum of 5 distinct 5th powers is 144^5 is 0^5 + 27^5 + 84^5 + 110^5 + 133^5 = 144^5, which is not the sum of 5 positive n-th powers).
Conjecture: a(6) = -1. (End)

			For n = 2: 625 = 25^2 = 7^2 + 24^2 = 15^2 + 20^2.
For n = 3: 157464 = 54^3 = 6^3 + 36^3 + 48^3 = 12^3 + 19^3 + 53^3 = 27^3 + 36^3 + 45^3.

Cf. A007666, A025303, A025401, A030052, A039664, A063923, A146756, A230718, A331675.

cp's OEIS Frontend

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

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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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A376289 Values k for primitive solutions to k^5 + a^5 + b^5 = c^5 + d^5 + e^5 with k >= a >= b >= 0 and k > c >= d >= e >= 0, repetitions allowed.

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A376914 Values k for primitive solutions to k^5 + a^5 + b^5 + c^5 = d^5 + e^5 with k >= a >= b >= c > 0 and d >= e >= 0, repetitions allowed.

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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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