A308944 - OEIS

%I #7 Jul 02 2019 09:10:12

%S 1,1,3,4,125,9,16807,1024,59049,15625,2357947691,5184,1792160394037,

%T 282475249,474609375,17179869184,2862423051509815793,3486784401,

%U 5480386857784802185939,250000000000,10382917022245341,5559917313492231481,39471584120695485887249589623

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

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

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

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

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

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

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

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

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

%t Table[Product[LCM[n, k]/(k GCD[n, k]), {k, 1, n}], {n, 1, 23}]

%t Table[Product[d^(EulerPhi[d] - EulerPhi[n/d]), {d, Divisors[n]}], {n, 1, 23}]

%o (PARI) a(n) = prod(k=1, n, lcm(n, k)/(k*gcd(n, k))); \\ _Michel Marcus_, Jul 02 2019

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

%K nonn

%O 1,3

%A _Ilya Gutkovskiy_, Jul 01 2019