%I #10 Jun 17 2016 00:31:08
%S 21,25,28,32,33,37,38,42,45,51,52,53,59,60,66,69,73,77,81,83,84,89,90,
%T 91,96,98,101,105,107,109
%N Numbers n such that n^k is of the form a^2 + b^3 + c^4 for all k > 0 (a, b, c > 0).
%C 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.
%C 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).
%e 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, ...
%e 441 is a term because 441 = 21^2.
%Y Cf. A123053.
%K nonn,more
%O 1,1
%A _Altug Alkan_, Jun 12 2016
%E a(2)-a(30) from _Giovanni Resta_, Jun 12 2016