A160377 - OEIS

%I #26 Oct 07 2021 02:00:05

%S 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,

%T 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,

%U 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

%N Phi-torial of n (A001783) modulo n.

%C Is a(n)<> 1 iff n in A033948, n>2? [_R. J. Mathar_, May 21 2009]

%C 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]

%H Amiram Eldar, <a href="/A160377/b160377.txt">Table of n, a(n) for n = 1..10000</a>

%H John B. Cosgrave and Karl Dilcher, <a href="http://www.emis.de/journals/INTEGERS/papers//i39/i39.Abstract.html"> Extensions of the Gauss-Wilson Theorem</a>, Integers: Electronic Journal of Combinatorial Number Theory, Vol. 8 (2008), Article #A39.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WilsonsTheorem.html">Wilson's Theorem</a>.

%F a(n) = A001783(n) mod n. - _R. J. Mathar_, May 21 2009

%F For n>2, a(n)=n-1 if A060594(n)=2; otherwise a(n)=1. - _Max Alekseyev_

%F a(n) = Gauss_factorial(n, n) modulo n. (Definition of the Gauss factorial in A216919.) - _Peter Luschny_, Oct 20 2012

%e Phi-torial of 12 equals 1*5*7*11=385 which leaves a remainder of 1 when divided by 12.

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

%p 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:

%p A001783 := proc(n) local c; mul(c,c= copr(n)) ; end:

%p A160377 := proc(n) A001783(n) mod n ; end: seq( A160377(n),n=1..100) ; # _R. J. Mathar_, May 21 2009

%p A160377 := proc(n) local k, r; r := 1:

%p for k to n do if igcd(n,k) = 1 then r := modp(r*k, n) fi od;

%p r end: seq( A160377(i), i=1..88); # _Peter Luschny_, Oct 20 2012

%t Table[nn = n; a = Select[Range[nn], CoprimeQ[#, nn] &];

%t Mod[Apply[Times, a], nn], {n, 1, 88}] (* _Geoffrey Critzer_, Jan 03 2015 *)

%o (Sage)

%o def A160377(n):

%o     r = 1

%o     for k in (1..n):

%o         if gcd(n, k) == 1: r = mod(r*k, n)

%o     return r

%o [A160377(n) for n in (1..88)]  # _Peter Luschny_, Oct 20 2012

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

%K nonn

%O 1,3

%A _J. M. Bergot_, May 11 2009

%E Edited and extended by _R. J. Mathar_ and _Max Alekseyev_, May 21 2009