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

1,3

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]

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.

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

Example

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.

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A280701 A064830 A355582 * A269165 A319655 A328018

Adjacent sequences: A160374 A160375 A160376 * A160378 A160379 A160380

Keyword

nonn

Author

J. M. Bergot, May 11 2009

Extensions

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

Status

approved