A308944 a(n) = Product_{k=1..n} lcm(n,k) / (k * gcd(n,k)).

1, 1, 3, 4, 125, 9, 16807, 1024, 59049, 15625, 2357947691, 5184, 1792160394037, 282475249, 474609375, 17179869184, 2862423051509815793, 3486784401, 5480386857784802185939, 250000000000, 10382917022245341, 5559917313492231481, 39471584120695485887249589623

Offset: 1

Author

Ilya Gutkovskiy, Jul 01 2019

Keywords

• nonn

Crossrefs

Cf. A000010, A051190, A056916, A067911, A071248, A119619, A127553.

Programs

Mathematica

Table[Product[LCM[n, k]/(k GCD[n, k]), {k, 1, n}], {n, 1, 23}]
Table[Product[d^(EulerPhi[d] - EulerPhi[n/d]), {d, Divisors[n]}], {n, 1, 23}]

PARI

a(n) = prod(k=1, n, lcm(n, k)/(k*gcd(n, k))); \\ Michel Marcus, Jul 02 2019

Formula

a(n) = Product_{d|n} d^(phi(d)-phi(n/d)).

a(n) = n^n / Product_{d|n} d^(2*phi(n/d)).

a(n) = n^(-n) * Product_{d|n} d^(2*phi(d)).

a(n) = n^n / Product_{k=1..n} gcd(n,k)^2.

a(n) = n^(-n) * Product_{k=1..n} lcm(n,k)^2/k^2.

a(n) = A127553(n)/n!.

a(n) = A056916(n)/A067911(n).

a(p) = p^(p-2), where p is a prime.