A224497 - OEIS

%I #12 Aug 02 2019 05:27:14

%S 1,1,1,1,2,2,12,12,96,288,5760,5760,829440,829440,46448640,2090188800,

%T 267544166400,267544166400,346737239654400,346737239654400,

%U 1109559166894080000,209706682542981120000,73816752255129354240000,73816752255129354240000

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

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

%C 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).

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

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

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

%p A224497 := n -> sqrt(iquo(n,2)!*mul(mul(igcd(k,i), i=1..k-1), k=1..n)):

%p seq(A224497(i), i = 0..23);

%o (Sage)

%o def A224497(n):

%o     R = 1;

%o     for p in primes(n):

%o         s = 0; t = 0

%o         r = n; u = n//2

%o         while r > 0 :

%o             r = r//p; u = u//p

%o             t += u; s += r*(r-1)

%o         R *= p^((t+s/2)/2)

%o     return R

%o [A224497(i) for i in (0..23)]

%Y Cf. A224479.

%K nonn

%O 0,5

%A _Peter Luschny_, Apr 08 2013