A056583 Solution to a(n)^(n^2/a(n)) = gcd(n^n, Product_{k<n} k^k) where a(n) and n^2/a(n) are integers, or 0 if no such integers exist.

1, 1, 0, 1, 12, 1, 16, 27, 20, 1, 12, 1, 28, 15, 16, 1, 18, 1, 20, 21, 44, 1, 24, 25, 52, 27, 28, 1, 30, 1, 32, 33, 68, 35, 36, 1, 76, 39, 40, 1, 42, 1, 44, 45, 92, 1, 48, 49, 50, 51, 52, 1, 54, 55, 56, 57, 116, 1, 60, 1, 124, 63, 64, 65, 66, 1, 68, 69, 70, 1, 72, 1, 148, 75, 76

Offset

2,5

Links

Antti Karttunen, Table of n, a(n) for n = 2..20000

Formula

a(2) = 1, a(4) = 0, a(8) = 16, a(9) = 27; if p an odd prime: a(p) = 1 and a(2p) = 4p; otherwise if n>1: a(n) = n. Apart from n = 4, a(n) = n^2/A056584(n) = A056582(n)^(1/A056584(n)).

Example

For n = 4, there are no integer solutions of a^(16/a) = 4, though there are two real solutions of about 1.099997 and 43.55926.

Prog

(PARI) A056583(n) = if(2==n, 1, if(4==n, 0, if(8==n, 16, if(9==n, 27, if(isprime(n), 1, if(!(n%2) && isprime(n/2), 2*n, n)))))); \\ Antti Karttunen, Jan 22 2025

Crossrefs

Cf. A000312, A002109, A056582, A056584.

Sequence in context: A392025 A070649 A255864 * A281823 A139724 A040155

Adjacent sequences: A056580 A056581 A056582 * A056584 A056585 A056586

Keyword

nonn

Author

Henry Bottomley, Jul 03 2000

Status

approved