A056582 -id:A056582 - cp's OEIS frontend

A056583 Solution to a(n)^(n^2/a(n)) = gcd(n^n, Product_{k

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

Henry Bottomley, Jul 03 2000

nonn

			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.

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

Cf. A000312, A002109, A056582, A056584.

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

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)).

A056584 Solution to (n^2/a(n))^a(n) = gcd(n^n, Product_{k

Original entry on oeis.org

4, 9, 0, 25, 3, 49, 4, 3, 5, 121, 12, 169, 7, 15, 16, 289, 18, 361, 20, 21, 11, 529, 24, 25, 13, 27, 28, 841, 30, 961, 32, 33, 17, 35, 36, 1369, 19, 39, 40, 1681, 42, 1849, 44, 45, 23, 2209, 48, 49, 50, 51, 52, 2809, 54, 55, 56, 57, 29, 3481, 60, 3721, 31, 63, 64, 65

Offset: 2

Henry Bottomley, Jul 03 2000

nonn

			For n = 4, there are no integer solutions of (16/a)^a = 4, though there are two real solutions of about 14.5454938 and 0.3673156945.

Cf. A000312, A002109, A056582, A056583.

a(2) = 4, a(4) = 0, a(8) = 4, a(9) = 3; if p an odd prime: a(p) = p^2 and a(2p) = p; otherwise if n>1: a(n) = n. Apart from n = 4, a(n) = n^2/A056583(n) = log(A056582(n))/log(A056583(n)).

cp's OEIS Frontend

A056583 Solution to a(n)^(n^2/a(n)) = gcd(n^n, Product_{k

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