A316537 - OEIS

%I #13 Jul 10 2018 18:43:23

%S 1,5,13,28,49,73,116,176,202,265,378,464,550,636,842,936,1041,1183,

%T 1486,1712,2082,2055,2120,3088,2114,3023,2503,4200,4238,4862,4902,

%U 4648,6564,5749,7434,7688,6331,8190,9880,11344,10172,12066,9378,13224,14168,11612

%N Number of cyclic subgroups of the group SL(2, Z(n)), counting conjugates as distinct.

%H Andrew Howroyd, <a href="/A316537/b316537.txt">Table of n, a(n) for n = 1..100</a>

%F a(n) = Sum_{k=1..A316563(n)} 1/phi(A316564(n, k)).

%e Case n=2: generators of the 5 cyclic groups are:

%e   [ 1 0 ]   [0 1]   [1 0]   [1 1]   [0 1]

%e   [ 0 1 ]   [1 0]   [1 1]   [0 1]   [1 1]

%o (GAP) Concatenation([1], List([2..10], n->Sum( Filtered( ConjugacyClassesSubgroups( SL(2, Integers mod n)), x->IsCyclic( Representative(x))), Size)));

%o (PARI)

%o MatOrder(M)={my(id=matid(#M), k=1, N=M); while(N<>id, k++;N=N*M); k}

%o a(n)={sum(a=0, n-1, sum(b=0, n-1, sum(c=0, n-1, sum(d=0, n-1, my(M=Mod([a, b; c, d], n)); if(matdet(M)==1, 1/eulerphi(MatOrder(M)))))))}

%Y Cf. A000056, A062314, A316536, A316553, A316560, A316563, A316564.

%K nonn

%O 1,2

%A _Andrew Howroyd_, Jul 06 2018