A385492 Numbers y such that there exists an integer 0 < x < y such that x^sigma(x) * y^sigma(y) = (x*y)^(x+y).

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 284, 293, 307, 311, 313

Offset: 1

Author

S. I. Dimitrov, Jun 30 2025

Keywords

• nonn

Comments

The numbers x and y form an IWGM-amicable pair (IWGM = Inverse Weighted Geometric Mean). See Dimitrov link.

Every amicable pair forms an IWGM-amicable pair, so the larger member of an amicable pair A002046 is a term of this sequence. In addition, every pair of the form (1, p), where p is a prime, also forms an IWGM-amicable pair, so every term of A000040 is a term of this sequence.

Most likely, there are no other solutions except those whose second component belongs to A000040 or A002046.

Examples

			(220, 284) is such a pair because 220^sigma(220)*284^sigma(284) = (220*284)^(220+284).
		

Links

• Robert Israel, Table of n, a(n) for n = 1..1000
• S. I. Dimitrov, Generalizations of amicable numbers, arXiv:2408.07387 [math.NT], 2024.

Crossrefs

Cf. A000203, A000040, A002046, A385186.

Programs

Maple

S:= map(numtheory:-sigma, [$1..1000]):
select(y -> ormap(x -> x^(S[x]-x-y) * y^(S[y]-x-y) = 1, [$1..y-1]), [$1..1000]); # Robert Israel, Jul 30 2025

PARI

isok(y) = my(z=y^sigma(y)); for (x=1, y-1, if (z*x^sigma(x) == (x*y)^(x+y), return(1));); \\ Michel Marcus, Jul 02 2025