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A070683 Smallest m in range 1..phi(2n+1) such that 12^m == 1 mod 2n+1, or 0 if no such number exists.

%I A070683 #11 Jun 04 2020 06:36:51
%S A070683 0,0,4,6,0,1,2,0,16,6,0,11,20,0,4,30,0,12,9,0,40,42,0,23,42,0,52,4,0,
%T A070683 29,15,0,4,66,0,35,36,0,6,26,0,41,16,0,8,6,0,12,16,0,100,102,0,53,54,
%U A070683 0,112,44,0,48,11,0,100,126,0,65,6,0,136,138,0,2,4,0,148
%N A070683 Smallest m in range 1..phi(2n+1) such that 12^m == 1 mod 2n+1, or 0 if no such number exists.
%C A070683 a(n)=2*n if 2*n+1 is in A019340, otherwise a(n)<2*n. - _Robert Israel_, Apr 17 2019
%H A070683 Robert Israel, <a href="/A070683/b070683.txt">Table of n, a(n) for n = 0..10000</a>
%p A070683 f:= proc(n)
%p A070683   if n mod 3 = 1 then 0 else numtheory:-order(12,2*n+1) fi
%p A070683 end proc:
%p A070683 0, seq(f(n),n=1..100); # _Robert Israel_, Apr 16 2019
%t A070683 a[n_] := Module[{s}, s = SelectFirst[Range[EulerPhi[2n+1]], PowerMod[12, #, 2n+1] == 1&]; If[s === Missing["NotFound"], 0, s]];
%t A070683 a /@ Range[0, 100] (* _Jean-François Alcover_, Jun 04 2020 *)
%Y A070683 Cf. A070667-A070675, A002326, A019340, A070676, A053447, A070677, A070678, A053451, A070679, A070682, A070680, A070681.
%K A070683 nonn,look
%O A070683 0,3
%A A070683 _N. J. A. Sloane_ and _Amarnath Murthy_, May 08 2002