%I A023136
%S A023136 1,1,3,1,3,3,3,1,5,3,3,3,3,3,9,1,5,5,3,3,9,3,3,3,5,3,7,3,3,9,7,1,9,5,9,
               5,
%T A023136 3,3,9,3,5,9,7,3,15,3,3,3,5,5,15,3,3,7,9,3,9,3,3,9,3,7,23,1,13,9,3,5,9,
               9,
%U A023136 3,5,9,3,15,3,9,9,3,3,9,5,3,9,23,7,9,3,9,15,17,3,21,3,9,3,5,5,15,5,3,15
%N A023136 Number of cycles of function f(x) = 4x mod n.
%H A023136 T. D. Noe, <a href="b023136.txt">Table of n, a(n) for n=1..10000</a>
%F A023136 a(n) = Sum_{d|m} phi(d)/ord(4, d), where m is n with all factors of 2 
               removed. The formula was developed by extending the ideas in A000374 
               to composite multipliers. - T. D. Noe (noe(AT)sspectra.com), Apr 
               21 2003
%F A023136 Mobius transform of A133702: (1, 2, 4, 3, 4, 8, 4, 4, 9, 8,...). = Row 
               sums of triangle A133703. - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Sep 21 2007
%e A023136 a(9) = 5 because the function 4x mod 9 has the five cycles (0),(3),(6),
               (1,4,7),(2,8,5).
%t A023136 CountFactors[p_, n_] := Module[{sum=0, m=n, d, f, i, ps, j}, ps=Transpose[FactorInteger[p]][[1]]; 
               Do[While[Mod[m, ps[[j]]]==0, m/=ps[[j]]], {j, Length[ps]}]; d=Divisors[m]; 
               Do[f=d[[i]]; sum+=EulerPhi[f]/MultiplicativeOrder[p, f], {i, Length[d]}]; 
               sum]; Table[CountFactors[4, n], {n, 100}]
%Y A023136 Cf. A000374, A023135-A023142.
%Y A023136 Cf. A133703, A133702.
%Y A023136 Sequence in context: A105595 A072219 A059789 this_sequence A152774 A068074 
               A063195
%Y A023136 Adjacent sequences: A023133 A023134 A023135 this_sequence A023137 A023138 
               A023139
%K A023136 nonn
%O A023136 1,3
%A A023136 David W. Wilson (davidwwilson(AT)comcast.net)