A232983 - OEIS

%I #14 Feb 24 2022 02:06:26

%S 1,1,2,6,24,120,720,720,5760,51840,518400,5702400,68428800,889574400,

%T 889574400,13343616000,213497856000,3629463552000,65330343936000,

%U 1241276534784000,24825530695680000,24825530695680000,546161675304960000,12561718532014080000,301481244768337920000,7537031119208448000000

%N The Gauss factorial n_7!.

%C The Gauss factorial n_k! is defined to be Product_{1<=j<=n, gcd(j,k)=1} j.

%H J. B. Cosgrave and K. Dilcher, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.118.09.812">An introduction to Gauss factorials</a>, Amer. Math. Monthly, 118 (2011), 810-828.

%H J. B. Cosgrave and K. Dilcher, <a href="http://dx.doi.org/10.1007/s10474-013-0357-1">The Gauss-Wilson theorem for quarter-intervals</a>, Acta Mathematica Hungarica, Sept. 2013.

%p Gf:=proc(N,n) local j,k; k:=1;

%p for j from 1 to N do if gcd(j,n)=1 then k:=j*k; fi; od; k; end;

%p f:=n->[seq(Gf(N,n),N=0..40)];

%p f(7);

%o (Magma) k:=7; [IsZero(n) select 1 else &*[j: j in [1..n] | IsOne(GCD(j,k))]: n in [0..30]]; // _Bruno Berselli_, Dec 10 2013

%Y The Gauss factorials n_1!, n_2!, n_3!, n_5!, n_6!, n_7!, n_10!, n_11! are A000142, A055634, A232980-A232985 respectively.

%K nonn

%O 0,3

%A _N. J. A. Sloane_, Dec 08 2013