A373867 Perfect powers of the form x^y + y^x, where x > 1 and y > 1.

8, 32, 100, 512, 33554432, 36893488147419103232, 2923003274661805836407369665432566039311865085952, 78804012392788958424558080200287227610159478540930893335896586808491443542994421222828532509769831281613255980613632

Offset: 1

Author

Gonzalo Martínez, Jun 21 2024

Keywords

• nonn

Comments

Subsequence of A076980: a(n) is a Leyland number that is a perfect power. The condition that x > 1 and y > 1 is necessary, otherwise every perfect power would belong to this sequence, since m^n = (m^n-1)^1 + 1^(m^n-1).

If x = y = 2^k, then x^y + y^x = 2^(k*2^k + 1) belongs to this sequence for all k > 0, and (k*2^k + 1) is the k-th Cullen number. That is, 2^A002064(k) is a term, with k > 0, from which it follows that this sequence has infinitely many terms.

Conjecture: 32 and 100 are the only terms for which x != y: 2^4 + 4^2 = 2^5 = 32 and 2^6 + 6^2 = 10^2 = 100.

Examples

			100 is a term, because 100 = 10^2 and 100 = 2^6 + 6^2.
		

Crossrefs

Cf. A001597, A002064, A076980.