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A232981 The Gauss factorial n_5!.

%I A232981 #21 Feb 24 2022 02:06:22
%S A232981 1,1,2,6,24,24,144,1008,8064,72576,72576,798336,9580032,124540416,
%T A232981 1743565824,1743565824,27897053184,474249904128,8536498274304,
%U A232981 162193467211776,162193467211776,3406062811447296,74933381851840512,1723467782592331776,41363226782215962624,41363226782215962624
%N A232981 The Gauss factorial n_5!.
%C A232981 The Gauss factorial n_k! is defined to be Product_{1<=j<=n, gcd(j,k)=1} j.
%H A232981 Robert Israel, <a href="/A232981/b232981.txt">Table of n, a(n) for n = 0..542</a>
%H A232981 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 A232981 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.
%F A232981 From _Robert Israel_, Mar 06 2017: (Start)
%F A232981 a(n) = a(n-1) if 5 | n; otherwise n*a(n-1).
%F A232981 a(n) = n!/(5^floor(n/5)*floor(n/5)!). (End)
%p A232981 Gf:=proc(N,n) local j,k; k:=1;
%p A232981 for j from 1 to N do if gcd(j,n)=1 then k:=j*k; fi; od; k; end;
%p A232981 f:=n->[seq(Gf(N,n),N=0..40)];
%p A232981 f(5);
%t A232981 Table[n!/(5^#*#!) &@ Floor[n/5], {n, 0, 25}] (* _Michael De Vlieger_, Mar 06 2017 *)
%o A232981 (Magma) k:=5; [IsZero(n) select 1 else &*[j: j in [1..n] | IsOne(GCD(j,k))]: n in [0..30]]; // _Bruno Berselli_, Dec 10 2013
%Y A232981 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 A232981 nonn
%O A232981 0,3
%A A232981 _N. J. A. Sloane_, Dec 08 2013