A181832 - OEIS

%I #31 Mar 23 2016 09:04:26

%S 1,1,1,1,1,3,1,20,15,35,7,36288,35,277200,1485,4576,9009,20432412000,

%T 5005,1097800704000,459459,5912192,2834325,2322315553259520000,

%U 1616615,124672148625024,4865140665

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

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

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

%C For 0 we have the empty product, giving 1. - _Daniel Forgues_, Aug 03 2012

%C From _Robert G. Wilson v_, Aug 04 2012: (Start)

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

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

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

%C 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)

%H Robert G. Wilson v, <a href="/A181832/b181832.txt">Table of n, a(n) for n = 0..1000</a>

%H Peter Luschny, <a href="http://www.oeis.org/wiki/User:Peter_Luschny/StrongCoprimality">Strong coprimality</a>.

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

%p with(numtheory):

%p StrongCoprimes := n -> select(k->igcd(k,n)=1,{$1..n}) minus divisors(n-1):

%p A181832 := proc(n) local i; mul(i,i=StrongCoprimes(n)) end:

%p coprimorial := proc(n) local i; mul(i,i=select(k->igcd(k,n)=1,[$1..n])) end:

%p divisorial  := proc(n) local i; mul(i,i=divisors(n)) end:

%p A181832a := n -> `if`(n=0,1,coprimorial(n)/divisorial(n-1)):

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

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

%K nonn

%O 0,6

%A _Peter Luschny_, Nov 17 2010