A280821 Partial products of A001783.

1, 1, 2, 6, 144, 720, 518400, 54432000, 121927680000, 23044331520000, 83623270219776000000, 32194959034613760000000, 15421436889514446422016000000000, 297710839152076388177018880000000000, 267015660792140704250415525396480000000000

Offset: 1

Author

Jaroslav Krizek, Jan 11 2017

Keywords

• nonn

Comments

A001783(n) = the product of totatives of n.

Links

• Project Euler, Problem 754: Product of Gauss Factorials.

Crossrefs

Cf. A000178, A001783, A280820, A281027.

Programs

Magma

[&*[&*[h: h in [1..k] | GCD(h,k) eq 1]: k in [1..n]]: n in [1..100]];

Mathematica

FoldList[#1 #2 &, Table[Times @@ Select[Range@ n, CoprimeQ[n, #] &], {n, 15}]] (* Michael De Vlieger, Jan 11 2017 *)
SetAttributes[Phitorial,{Listable}]
Phitorial[n_]:=n^EulerPhi[n]*Times@@((Factorial[#]/#^#)^MoebiusMu[n/#]&/@Divisors[n])
FoldList[Times,Phitorial[Range[20]]] (* Peter Cullen Burbery, Jul 14 2023 *)

PARI

f(n) = prod(k=2, n-1, k^(gcd(k, n)==1)); \\ A001783
a(n) = prod(i=1, n, f(i)); \\ Michel Marcus, Jul 14 2023

Formula

a(n) = Product_{i=1..n} A001783(i).

a(n) = A000178(n)/A281027(n). - Amiram Eldar, Aug 16 2025