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

Author

Altug Alkan, Jun 12 2016

Keywords

• nonn
• more

Comments

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).

Examples

			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.
		

Crossrefs

Cf. A123053.

Extensions

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