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%I #26 Nov 01 2019 09:35:56
%S 1,1,1,1,2,1,6,3,8,3,120,5,720,15,56,105,40320,35,362880,189,3200,945,
%T 39916800,385,9580032,10395,3203200,19305,87178291200,1001,
%U 1307674368000,2027025,65228800,2027025,4839284736,85085,6402373705728000,34459425,17827532800
%N a(n) = Product_{1<=k<=n/2, gcd(k,n)=1} k.
%C a(n) divides A001783(n). - _M. F. Hasler_, Jul 23 2011
%H Alois P. Heinz, <a href="/A124441/b124441.txt">Table of n, a(n) for n = 1..800</a>
%H J. B. Cosgrave and K. Dilcher, <a href="http://www.emis.de/journals/INTEGERS/papers/i39/i39.Abstract.html">Extensions of the Gauss-Wilson Theorem</a>, Integers: Electronic Journal of Combinatorial Number Theory, 8 (2008)
%F a(n) = A001783(n)/A124442(n). - _M. F. Hasler_, Jul 23 2011
%e The positive integers which are <= 9/2 and which are coprime to 9 are 1, 2 and 4. So a(9) = 1*2*4 = 8.
%p a:=proc(n) local b,k: b:=1: for k from 1 to floor(n/2) do if gcd(k,n)=1 then b:=b*k else b:=b fi od: b; end: seq(a(n),n=1..41); # _Emeric Deutsch_, Nov 03 2006
%t f[n_] := Times @@ Select[Range[Floor[n/2]], GCD[ #, n] == 1 &];Table[f[n], {n, 36}] (* _Ray Chandler_, Nov 12 2006 *)
%o (PARI) A124441(n)=prod(k=2, n\2, k^(gcd(k, n)==1)) \\ _M. F. Hasler_, Jul 23 2011
%o (Sage)
%o def Gauss_factorial(N, n): return mul(j for j in (1..N) if gcd(j, n) == 1)
%o def A124441(n): return Gauss_factorial(n//2, n)
%o [A124441(n) for n in (1..36)] # _Peter Luschny_, Oct 01 2012
%Y Cf. A124442, A001783.
%K nonn
%O 1,5
%A _Leroy Quet_, Nov 01 2006
%E More terms from _Emeric Deutsch_, Nov 03 2006
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