A232981 The Gauss factorial n_5!.

1, 1, 2, 6, 24, 24, 144, 1008, 8064, 72576, 72576, 798336, 9580032, 124540416, 1743565824, 1743565824, 27897053184, 474249904128, 8536498274304, 162193467211776, 162193467211776, 3406062811447296, 74933381851840512, 1723467782592331776, 41363226782215962624, 41363226782215962624

Offset: 0

Author

N. J. A. Sloane, Dec 08 2013

Keywords

• nonn

Comments

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

Links

• Robert Israel, Table of n, a(n) for n = 0..542
• J. B. Cosgrave and K. Dilcher, An introduction to Gauss factorials, Amer. Math. Monthly, 118 (2011), 810-828.
• J. B. Cosgrave and K. Dilcher, The Gauss-Wilson theorem for quarter-intervals, Acta Mathematica Hungarica, Sept. 2013.

Crossrefs

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.

Programs

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

Maple

Gf:=proc(N,n) local j,k; k:=1;
for j from 1 to N do if gcd(j,n)=1 then k:=j*k; fi; od; k; end;
f:=n->[seq(Gf(N,n),N=0..40)];
f(5);

Mathematica

Table[n!/(5^#*#!) &@ Floor[n/5], {n, 0, 25}] (* Michael De Vlieger, Mar 06 2017 *)

Formula

From Robert Israel, Mar 06 2017: (Start)

a(n) = a(n-1) if 5 | n; otherwise n*a(n-1).

a(n) = n!/(5^floor(n/5)*floor(n/5)!). (End)