A146756 -id:A146756 - cp's OEIS frontend

A360306 a(n) is the smallest positive integer which can be represented as the sum of n distinct nonzero fourth powers in exactly n ways, or -1 if no such integer exists.

1, 635318657, 811538, 300834, 185299, 138595, 143651, 154292, 197748, 225733, 291269, 374790, 474071, 586056, 715192, 857513, 1057689, 1330554, 1602250, 1919146, 2329547, 2786843, 3246204, 3899260, 4642700, 5378141, 6377822, 7342638, 8527103, 9787839, 11241455, 12978656

Offset: 1

Ilya Gutkovskiy, Feb 02 2023

nonn

			For n = 2: 635318657 = 59^4 + 158^4 = 133^4 + 134^4.
For n = 3: 811538 = 4^4 + 23^4 + 27^4 = 7^4 + 21^4 + 28^4 = 12^4 + 17^4 + 29^4.
For n = 4: 300834 = 1^4 + 4^4 + 12^4 + 23^4 = 1^4 + 16^4 + 18^4 + 19^4 = 3^4 + 6^4 + 18^4 + 21^4 = 7^4 + 14^4 + 16^4 + 21^4.

Bert Dobbelaere, Table of n, a(n) for n = 1..79
Bert Dobbelaere, C++ program
Eric Weisstein's World of Mathematics, Biquadratic Number

Cf. A000583, A146756, A350241, A350270.

a(18)-a(19) from Michael S. Branicky, Feb 04 2023

More terms from Bert Dobbelaere, Feb 11 2023

A374269 a(n) is the smallest number which can be represented as the sum of n distinct positive n-th powers in exactly 4 ways, or -1 if no such number exists.

Original entry on oeis.org

-1, 1105, 13896, 300834, 1386406515, 2063792939

Offset: 1

Ilya Gutkovskiy, Jul 02 2024

			a(2) = 1105 = 4^2 + 33^2 = 9^2 + 32^2 = 12^2 + 31^2 = 23^2 + 24^2.
a(3) = 13896 = 1^3 + 12^3 + 23^3 = 2^3 + 4^3 + 24^3 = 4^3 + 18^3 + 20^3 = 9^3 + 10^3 + 23^3.

Cf. A025305, A146756, A374228, A374256, A374257.

a(6) from Michael S. Branicky, Jul 09 2024

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

A360306 a(n) is the smallest positive integer which can be represented as the sum of n distinct nonzero fourth powers in exactly n ways, or -1 if no such integer exists.

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A374269 a(n) is the smallest number which can be represented as the sum of n distinct positive n-th powers in exactly 4 ways, or -1 if no such number exists.

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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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