A160377 Phi-torial of n (A001783) modulo n.

0, 1, 2, 3, 4, 5, 6, 1, 8, 9, 10, 1, 12, 13, 1, 1, 16, 17, 18, 1, 1, 21, 22, 1, 24, 25, 26, 1, 28, 1, 30, 1, 1, 33, 1, 1, 36, 37, 1, 1, 40, 1, 42, 1, 1, 45, 46, 1, 48, 49, 1, 1, 52, 53, 1, 1, 1, 57, 58, 1, 60, 61, 1, 1, 1, 1, 66, 1, 1, 1, 70, 1, 72, 73, 1, 1, 1, 1, 78, 1, 80, 81, 82, 1, 1, 85, 1, 1

Offset: 1

Author

J. M. Bergot, May 11 2009

Keywords

• nonn

Comments

Is a(n)<> 1 iff n in A033948, n>2? [R. J. Mathar, May 21 2009]

Same as A103131, except there -1 appears instead of n-1. By Gauss's generalization of Wilson's theorem, a(n)=-1 means n has a primitive root (n in A033948) and a(n)=1 means n has no primitive root (n in A033949). [T. D. Noe, May 21 2009]

Examples

			Phi-torial of 12 equals 1*5*7*11=385 which leaves a remainder of 1 when divided by 12.
Phi-torial of 14 equals 1*3*5*9*11*13=19305 which leaves a remainder of 13 when divided by 14.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• John B. Cosgrave and Karl Dilcher, Extensions of the Gauss-Wilson Theorem, Integers: Electronic Journal of Combinatorial Number Theory, Vol. 8 (2008), Article #A39.
• Eric Weisstein's World of Mathematics, Wilson's Theorem.

Crossrefs

Cf. A124740 (one of just four listing "product of coprimes").

Programs

Maple

copr := proc(n) local a,k ; a := {1} ; for k from 2 to n-1 do if gcd(k,n) = 1 then a := a union {k} ; fi; od: a ; end:
A001783 := proc(n) local c; mul(c,c= copr(n)) ; end:
A160377 := proc(n) A001783(n) mod n ; end: seq( A160377(n),n=1..100) ; # R. J. Mathar, May 21 2009
A160377 := proc(n) local k, r; r := 1:
for k to n do if igcd(n,k) = 1 then r := modp(r*k, n) fi od;
r end: seq( A160377(i), i=1..88); # Peter Luschny, Oct 20 2012

Mathematica

Table[nn = n; a = Select[Range[nn], CoprimeQ[#, nn] &];
Mod[Apply[Times, a], nn], {n, 1, 88}] (* Geoffrey Critzer, Jan 03 2015 *)

Sage

def A160377(n):
    r = 1
    for k in (1..n):
        if gcd(n, k) == 1: r = mod(r*k, n)
    return r
[A160377(n) for n in (1..88)]  # Peter Luschny, Oct 20 2012

Formula

a(n) = A001783(n) mod n. - R. J. Mathar, May 21 2009

For n>2, a(n)=n-1 if A060594(n)=2; otherwise a(n)=1. - Max Alekseyev

a(n) = Gauss_factorial(n, n) modulo n. (Definition of the Gauss factorial in A216919.) - Peter Luschny, Oct 20 2012

Extensions

Edited and extended by R. J. Mathar and Max Alekseyev, May 21 2009