A331675 -id:A331675 - cp's OEIS frontend

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.

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

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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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.

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