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A103491 Multiplicative suborder of 5 (mod n) = sord(5, n).

%I A103491 #24 Feb 16 2025 08:32:56
%S A103491 0,0,1,1,1,0,1,3,2,3,0,5,2,2,3,0,4,8,3,9,0,3,5,11,2,0,2,9,6,7,0,3,8,
%T A103491 10,8,0,6,18,9,4,0,10,3,21,5,0,11,23,4,21,0,16,4,26,9,0,6,18,7,29,0,
%U A103491 15,3,3,16,0,10,11,16,11,0,5,6,36,18,0,9,30,4,39,0,27,10,41,6,0,21,7,10,22,0
%N A103491 Multiplicative suborder of 5 (mod n) = sord(5, n).
%C A103491 a(n) is minimum e for which 5^e = +/-1 mod n, or zero if no e exists.
%C A103491 For n > 2, a(n) <= (n-1)/2, with equality if (but not only if) n is in A019335. - _Robert Israel_, Mar 20 2020
%D A103491 H. Cohen, Course in Computational Algebraic Number Theory, Springer, 1993, p. 25, Algorithm 1.4.3
%H A103491 Robert Israel, <a href="/A103491/b103491.txt">Table of n, a(n) for n = 0..10000</a>
%H A103491 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MultiplicativeOrder.html">Multiplicative Order</a>.
%H A103491 S. Wolfram, <a href="http://www.stephenwolfram.com/publications/articles/ca/84-properties/9/text.html">Algebraic Properties of Cellular Automata (1984)</a>, Appendix B.
%H A103491 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SuborderFunction.html">Suborder Function</a>
%p A103491 f:= proc(n) local x;
%p A103491   if n mod 5 = 0 then return 0 fi;
%p A103491   x:= numtheory:-mlog(-1,5,n);
%p A103491   if x <> FAIL then x else numtheory:-order(5,n) fi
%p A103491 end proc:
%p A103491 f(1):= 0:
%p A103491 map(f, [$0..100]); # _Robert Israel_, Mar 20 2020
%t A103491 Suborder[k_, n_] := If[n > 1 && GCD[k, n] == 1, Min[MultiplicativeOrder[k, n, {-1, 1}]], 0];
%t A103491 a[n_] := Suborder[5, n];
%t A103491 a /@ Range[0, 100] (* _Jean-François Alcover_, Mar 21 2020, after _T. D. Noe_ in A003558 *)
%Y A103491 Cf. A019335.
%K A103491 easy,nonn
%O A103491 0,8
%A A103491 _Harry J. Smith_, Feb 08 2005