%I A023140
%S A023140 1,1,2,1,2,2,7,1,5,2,2,2,4,7,5,1,3,5,4,2,14,2,3,2,3,4,8,7,2,5,7,1,5,3,
               14,
%T A023140 5,4,4,11,2,3,14,4,2,14,3,3,2,13,3,8,4,2,8,5,7,11,2,2,5,4,7,35,1,17,5,
               4,
%U A023140 3,6,14,3,5,25,4,8,4,14,11,7,2,11,3,2,14,12,4,5,2,9,14,28,3,14,3,11,2,
               7
%N A023140 Number of cycles of function f(x) = 8x mod n.
%H A023140 T. D. Noe, <a href="b023140.txt">Table of n, a(n) for n=1..10000</a>
%F A023140 a(n) = Sum_{d|m} phi(d)/ord(8, d), where m is n with all factors of 2 
               removed. - T. D. Noe (noe(AT)sspectra.com), Apr 21 2003
%e A023140 a(10) = 2 because the function 8x mod 10 has the two cycles (0),(2,6,
               8,4).
%t A023140 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[8, n], {n, 100}]
%Y A023140 Cf. A000374, A023135-A023142.
%Y A023140 Sequence in context: A000020 A077014 A093655 this_sequence A145859 A145863 
               A110775
%Y A023140 Adjacent sequences: A023137 A023138 A023139 this_sequence A023141 A023142 
               A023143
%K A023140 nonn
%O A023140 1,3
%A A023140 David W. Wilson (davidwwilson(AT)comcast.net)