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

%I A160377 #27 Feb 16 2025 08:33:10
%S A160377 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 A160377 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 A160377 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 A160377 Phi-torial of n (A001783) modulo n.
%C A160377 Is a(n)<> 1 iff n in A033948, n>2? [_R. J. Mathar_, May 21 2009]
%C A160377 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 A160377 Amiram Eldar, <a href="/A160377/b160377.txt">Table of n, a(n) for n = 1..10000</a>
%H A160377 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 A160377 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WilsonsTheorem.html">Wilson's Theorem</a>.
%F A160377 a(n) = A001783(n) mod n. - _R. J. Mathar_, May 21 2009
%F A160377 For n>2, a(n)=n-1 if A060594(n)=2; otherwise a(n)=1. - _Max Alekseyev_
%F A160377 a(n) = Gauss_factorial(n, n) modulo n. (Definition of the Gauss factorial in A216919.) - _Peter Luschny_, Oct 20 2012
%e A160377 Phi-torial of 12 equals 1*5*7*11=385 which leaves a remainder of 1 when divided by 12.
%e A160377 Phi-torial of 14 equals 1*3*5*9*11*13=19305 which leaves a remainder of 13 when divided by 14.
%p A160377 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 A160377 A001783 := proc(n) local c; mul(c,c= copr(n)) ; end:
%p A160377 A160377 := proc(n) A001783(n) mod n ; end: seq( A160377(n),n=1..100) ; # _R. J. Mathar_, May 21 2009
%p A160377 A160377 := proc(n) local k, r; r := 1:
%p A160377 for k to n do if igcd(n,k) = 1 then r := modp(r*k, n) fi od;
%p A160377 r end: seq( A160377(i), i=1..88); # _Peter Luschny_, Oct 20 2012
%t A160377 Table[nn = n; a = Select[Range[nn], CoprimeQ[#, nn] &];
%t A160377 Mod[Apply[Times, a], nn], {n, 1, 88}] (* _Geoffrey Critzer_, Jan 03 2015 *)
%o A160377 (Sage)
%o A160377 def A160377(n):
%o A160377     r = 1
%o A160377     for k in (1..n):
%o A160377         if gcd(n, k) == 1: r = mod(r*k, n)
%o A160377     return r
%o A160377 [A160377(n) for n in (1..88)]  # _Peter Luschny_, Oct 20 2012
%Y A160377 Cf. A124740 (one of just four listing "product of coprimes").
%K A160377 nonn
%O A160377 1,3
%A A160377 _J. M. Bergot_, May 11 2009
%E A160377 Edited and extended by _R. J. Mathar_ and _Max Alekseyev_, May 21 2009