A066570 Product of numbers <= n that have a prime factor in common with n.

1, 2, 3, 8, 5, 144, 7, 384, 162, 19200, 11, 1244160, 13, 4515840, 1458000, 10321920, 17, 75246796800, 19, 278691840000, 1080203040, 899245670400, 23, 16686729658368000, 375000, 663152807116800, 7142567040, 209964381084057600, 29, 1229978843118305280000000

Offset: 1

Author

Amarnath Murthy, Dec 19 2001

Keywords

• nonn
• easy

Comments

Empty product, 1, for n = 1.

a(p) = p if p is a prime.

Examples

			a(7) = 7, a(9) = 3*6*9 = 162.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A001783, A216919.

Programs

Maple

A066570 := proc(n) local i; mul(i,i=remove(k->igcd(n,k)=1,[$1..n])) end: # Peter Luschny, Oct 11 2011

Mathematica

Table[Times @@ Select[Range[2, n], GCD[#, n] > 1 &], {n, 30}] (* T. D. Noe, Oct 04 2012 *)

PARI

a(n) = prod(k=1, n, if (gcd(k, n) != 1, k, 1)); \\ Michel Marcus, Nov 02 2017

Sage

def Gauss_factorial(N, n): return mul(j for j in (1..N) if gcd(j, n) == 1)
def A066570(n): return Gauss_factorial(n, 1)/Gauss_factorial(n, n)
[A066570(n) for n in (1..30)] # Peter Luschny, Oct 02 2012

Formula

a(n) = n!/A001783(n).

a(n) = Gauss_factorial(n, 1)/Gauss_factorial(n, n) (see A216919). - Peter Luschny, Oct 02 2012