A274356 Numbers n such that n^k is of the form (a^2 + b^4)/2 for all k > 0 (a, b > 0).

1, 5, 16, 25, 41, 80, 81, 125, 256, 400, 405, 425, 625, 656, 841, 1225, 1280, 1296, 1681, 2000, 2025, 2401, 3125, 3321, 3721, 4096, 6400, 6480, 6561, 6800, 8281, 8381, 10000, 10125, 10496, 12005, 13456, 14161, 14641, 15625, 19600, 20480, 20736, 25625

Offset: 1

Author

Altug Alkan, Jun 18 2016

Keywords

• nonn

Comments

Numbers n such that n^k is the average of a nonzero square and a nonzero fourth power for all k > 0.

If n^k = (a^2 + b^4)/2, then n^(k+4) = ((n^2*a)^2 + (n*b)^4)/2. So this sequence lists numbers n such that 2*n, 2*n^2 and 2*n^3 are in A111925.

If n is in this sequence, then n^t is also in this sequence for all t > 1. So in this sequence there are infinitely many prime powers.

There are infinitely many (x, y) pairs in this sequence where x and y are distinct terms of this sequence such that x*y is also in this sequence.

Examples

			5 is a term because 5 = (3^2 + 1^4)/2, 5^2 = (7^2 + 1^4)/2, 5^3 = (13^2 + 3^4)/2, 5^4 = ((5^2)^2 + 5^4)/2.
		

Crossrefs

Cf. A111925.

Programs

PARI

isA111925(n)=for(b=1, sqrtnint(n-1, 4), if(issquare(n-b^4), return(1))); 0;
lista(nn)=for(n=1, nn, if(isA111925(2*n) && isA111925(2*n^2) && isA111925(2*n^3), print1(n, ", ")));