A224497 a(n) = sqrt(floor(n/2)! * Product_{k=1..n} Product_{i=1..k-1} gcd(k,i)).

1, 1, 1, 1, 2, 2, 12, 12, 96, 288, 5760, 5760, 829440, 829440, 46448640, 2090188800, 267544166400, 267544166400, 346737239654400, 346737239654400, 1109559166894080000, 209706682542981120000, 73816752255129354240000, 73816752255129354240000

Offset: 0

Author

Peter Luschny, Apr 08 2013

Keywords

• nonn

Comments

The order of the primes in the prime factorization of a(n) is given by

ord_{p}(a(n)) = (1/4)*Sum_{i>=1} floor(n/p^i)*(floor(n/p^i)-1) + (1/2)*Sum_{i>=1} floor(floor(n/2)/p^i).

For n > 1: a(n) = a(n-1) if and only if n is prime.

Crossrefs

Cf. A224479.

Programs

Maple

A224497 := n -> sqrt(iquo(n,2)!*mul(mul(igcd(k,i), i=1..k-1), k=1..n)):
seq(A224497(i), i = 0..23);

Sage

def A224497(n):
    R = 1;
    for p in primes(n):
        s = 0; t = 0
        r = n; u = n//2
        while r > 0 :
            r = r//p; u = u//p
            t += u; s += r*(r-1)
        R *= p^((t+s/2)/2)
    return R
[A224497(i) for i in (0..23)]

Formula

a(n) = sqrt(floor(n/2)! * A224479(n)).

A092287(n) = A056040(n) * a(n)^4.