A123053 -id:A123053 - cp's OEIS frontend

A274178 Numbers n such that n^k is of the form a^2 + b^3 + c^4 for all k > 0 (a, b, c > 0).

21, 25, 28, 32, 33, 37, 38, 42, 45, 51, 52, 53, 59, 60, 66, 69, 73, 77, 81, 83, 84, 89, 90, 91, 96, 98, 101, 105, 107, 109

Offset: 1

Altug Alkan, Jun 12 2016

If n is a term of this sequence, then n^t is also in this sequence for all t > 1. So sequence is infinite by definition.

If n^k = a^2 + b^3 + c^4, then n^(k+12) = (a*n^6)^2 + (b*n^4)^3 + (c*n^3)^4. So if n^k is in A123053 for all 1 <= k <= 12, then n^k is of the form a^2 + b^3 + c^4 for all k > 0 (a, b, c > 0).

			21 is a term because 21 = 2^2 + 1^3 + 2^4, 21^2 = 12^2 + 6^3 + 3^4, 21^3 = 1^2 + 19^3 + 7^4, 21^4 = 424^2 + 4^3 + 11^4, 21^5 = 458^2 + 116^3 + 39^4, 21^6 = 6345^2 + 135^3 + 81^4, 21^7 = 38062^2 + 46^3 + 137^4, 21^8 = 91728^2 + 2096^3 + 377^4, 21^9 = 887395^2 + 1795^3 + 179^4, 21^10 = 1541557^2 + 24271^3 + 277^4, 21^11 = 10833858^2 + 61526^3 + 197^4, 21^12 = 6063740^2 + 194156^3 + 465^4, 21^13 = 392733406^2 + 61520^3 + 345^4, ...
441 is a term because 441 = 21^2.

Cf. A123053.

a(2)-a(30) from Giovanni Resta, Jun 12 2016

A333841 Integers n such that n! = x^2 + y^3 + z^4 where x, y and z are nonnegative integers, is soluble.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24

Offset: 1

Altug Alkan, Aug 14 2020

			6! = 11^2+7^3+4^4; 8! = 192^2+15^3+3^4;  9! = 443^2+55^3+4^4; 10! = 1888^2+40^3+4^4; 11! = 5896^2+172^3+16^4, so 6, 8, 9, 10 and 11 are in the sequence. - _R. J. Mathar_, Dec 15 2020

Cf. A123053, A267414, A337046.

{k: k! in A123053}. - R. J. Mathar, Dec 15 2020

cp's OEIS Frontend

A274178 Numbers n such that n^k is of the form a^2 + b^3 + c^4 for all k > 0 (a, b, c > 0).

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