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%I #12 Feb 24 2022 02:05:01
%S 1,1,2,2,8,40,40,280,2240,2240,22400,246400,246400,3203200,44844800,
%T 44844800,717516800,12197785600,12197785600,231757926400,
%U 4635158528000,4635158528000,101973487616000,2345390215168000,2345390215168000,58634755379200000,1524503639859200000,1524503639859200000
%N The Gauss factorial n_3!.
%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(3);
%o (Magma) k:=3; [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
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