A181832 The product of the positive integers <= n that are strongly prime to n.

1, 1, 1, 1, 1, 3, 1, 20, 15, 35, 7, 36288, 35, 277200, 1485, 4576, 9009, 20432412000, 5005, 1097800704000, 459459, 5912192, 2834325, 2322315553259520000, 1616615, 124672148625024, 4865140665

Offset: 0

Author

Peter Luschny, Nov 17 2010

Keywords

• nonn

Comments

k is strongly prime to n iff k is relatively prime to n and k does not divide n-1.

a(n) = A001783(n) / A007955(n-1) if n > 0 and a(0) = 1.

For 0 we have the empty product, giving 1. - Daniel Forgues, Aug 03 2012

From Robert G. Wilson v, Aug 04 2012: (Start)

Records appear at positions 0, 5, 7, 9, 11, 13, 17, 19, 23, 29, 31, ....

Except for 0 and 9, all records appear at prime positions and beginning with the sixth term, are == 0 (mod 100).

There are some primes which are not records: 2, 3, 61, 73, 109, 151, 181, 193, 229, 241, 271, 313, 349, 421, 433, 463, ....

Anti-records appear at positions 6, 10, 12, 14, 15, 18, 20, 24, 30, 36, 42, 48, 60, 66, 70, 78, 84, 90, 96, ..., and their values are odd. (End)

Examples

			a(11) = 3 * 4 * 6 * 7 * 8 * 9 = 36288.
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 0..1000
• Peter Luschny, Strong coprimality.

Crossrefs

Cf. A181830, A181831, A181833, A181834, A181835, A181836, A001783, A007955.

Programs

Maple

with(numtheory):
StrongCoprimes := n -> select(k->igcd(k,n)=1,{$1..n}) minus divisors(n-1):
A181832 := proc(n) local i; mul(i,i=StrongCoprimes(n)) end:
coprimorial := proc(n) local i; mul(i,i=select(k->igcd(k,n)=1,[$1..n])) end:
divisorial  := proc(n) local i; mul(i,i=divisors(n)) end:
A181832a := n -> `if`(n=0,1,coprimorial(n)/divisorial(n-1)):

Mathematica

f[n_] := Times @@ Select[ Range@ n, GCD[#, n] == 1 && Mod[n - 1, #] != 0 &]; Array[f, 27, 0] (* Robert G. Wilson v, Aug 03 2012 *)