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 A103131 The product of the residues in [1,n] relatively prime to n, taken modulo n. 3
 0, 1, -1, -1, -1, -1, -1, 1, -1, -1, -1, 1, -1, -1, 1, 1, -1, -1, -1, 1, 1, -1, -1, 1, -1, -1, -1, 1, -1, 1, -1, 1, 1, -1, 1, 1, -1, -1, 1, 1, -1, 1, -1, 1, 1, -1, -1, 1, -1, -1, 1, 1, -1, -1, 1, 1, 1, -1, -1, 1, -1, -1, 1, 1, 1, 1, -1, 1, 1, 1, -1, 1, -1, -1, 1, 1, 1, 1, -1, 1, -1, -1, -1, 1, 1, -1, 1, 1, -1, 1, 1, 1, 1, -1, 1, 1, -1, -1, 1, 1, -1, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Old name was: Minimal residue (in absolute value) of A001783(n) (mod n). If the positive representation for integers modulo n is used this is A160377. Here the symmetric (or minimal) representation for the integers modulo n is used. From Gauss's generalization of Wilson's theorem (see Weisstein link) it follows that, for n>1, a(n) = -1 if and only if there exists a primitive root modulo n (cf. the Hardy and Wright reference, Theorem 129. p. 102). (Adapted from a comment by Vladimir Shevelev in A001783). - Peter Luschny, Oct 20 2012 REFERENCES G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth ed., Clarendon Press, Oxford, 2003, Theorem 129, p. 102. LINKS Antti Karttunen, Table of n, a(n) for n = 1..16385 J. B. Cosgrave, K. Dilcher, Extensions of the Gauss-Wilson Theorem, Integers: Electronic Journal of Combinatorial Number Theory, 8 (2008). Eric Weisstein's World of Mathematics, Wilson's theorem FORMULA For n>2, a(n)=-1 if A060594(n)=2, or equivalently if n is in A033948; otherwise a(n)=1. - Max Alekseyev, May 26 2009; edited by Peter Luschny, May 25 2017. a(n) = Gauss_factorial(n, n) modulo n. (Definition of the Gauss factorial in A216919.) - Peter Luschny, Oct 20 2012 For n > 2, a(n) = (-1)^A211487(n). (See Max Alekseyev's formula above.) - Antti Karttunen, Aug 22 2017 EXAMPLE The residues in [1, 22] relatively prime to 22 are [1, 3, 5, 7, 9, 13, 15, 17, 19, 21] and the product of those residues is -1 modulo 22. MAPLE A103131 := proc(n) local k, r; r := 1; for k to n do if igcd(n, k) = 1 then r := mods(r*k, n) fi od; r end: seq(A103131(i), i=1..102);   # Peter Luschny, Oct 20 2012 MATHEMATICA a[n_] := If[IntegerQ[PrimitiveRoot[n]], -1, 1]; a[1] = 0; a[2] = 1; Table[a[n], {n, 1, 102}] (* Jean-François Alcover, Nov 09 2012, after Max Alekseyev *) PROG (Sage) def A103131(n):     def smod(a, n): return a-n*ceil(a/n-1/2) if n != 0 else a     r = 1     for k in (1..n):         if gcd(n, k) == 1: r = smod(r*k, n)     return r [A103131(n) for n in (1..102)]  # Peter Luschny, Oct 20 2012 (PARI) A211487(n) = if(n%2, !!isprimepower(n), (n==2 || n==4 || (isprimepower(n/2, &n) && n>2))); \\ After Charles R Greathouse IV's code for A033948. A103131(n) = if(n<=2, n-1, (-1)^A211487(n)); \\ Antti Karttunen, Aug 22 2017 CROSSREFS Cf. A001783, A160377, A211487, A216919. Sequence in context: A226523 A070238 A062157 * A112347 A057427 A178334 Adjacent sequences:  A103128 A103129 A103130 * A103132 A103133 A103134 KEYWORD sign AUTHOR Eric W. Weisstein, Jan 23 2005 EXTENSIONS Definition rewritten by Max Alekseyev, May 26 2009 New name from Peter Luschny, Oct 20 2012 a(2) set to 1 by Peter Luschny, May 25 2017 STATUS approved

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Last modified November 30 12:24 EST 2020. Contains 338802 sequences. (Running on oeis4.)