A316553 -id:A316553 - cp's OEIS frontend

A316564 Triangle read by rows: T(n,k) is the number of elements of the group SL(2, Z(n)) with order k, 1 <= k <= A316563(n).

1, 1, 3, 2, 1, 1, 8, 6, 0, 8, 1, 7, 8, 24, 0, 8, 1, 1, 20, 30, 24, 20, 0, 0, 0, 24, 1, 7, 26, 24, 0, 74, 0, 0, 0, 0, 0, 12, 1, 1, 56, 42, 0, 56, 48, 84, 0, 0, 0, 0, 0, 48, 1, 15, 32, 144, 0, 96, 0, 96, 1, 1, 98, 54, 0, 98, 0, 0, 144, 0, 0, 108, 0, 0, 0, 0, 0, 144

Offset: 1

Andrew Howroyd, Jul 06 2018

For coprime p,q the group SL(p*q, Z(n)) is isomorphic to the direct product of the two groups SL(p, Z(n)) and SL(q, Z(n)).

			Triangle begins:
  1;
  1,  3,  2;
  1,  1,  8, 6, 0, 8;
  1,  7,  8, 24, 0, 8;
  1,  1, 20, 30, 24, 20, 0, 0, 0, 24;
  1,  7, 26, 24, 0, 74, 0, 0, 0, 0, 0, 12;
  1,  1, 56, 42, 0, 56, 48, 84, 0, 0, 0, 0, 0, 48;
  1, 15, 32, 144, 0, 96, 0, 96;
  1,  1, 98, 54, 0, 98, 0, 0, 144, 0, 0, 108, 0, 0, 0, 0, 0, 144;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..3478 (first 60 rows)

Column 2 is A316553.
Row sums are A000056.
Cf. A316537, A316566, A316586.

MatOrder(M)={my(id=matid(#M), k=1, N=M); while(N<>id, k++;N=N*M); k}
row(n)={my(L=List()); for(a=0, n-1, for(b=0, n-1, for(c=0, n-1, for(d=0, n-1, my(M=Mod([a, b; c, d], n)); if(matdet(M)==1, my(t=MatOrder(M)); while(#L

T(p*q,k) = Sum_{i>0, j>0, k=lcm(i, j)} T(p, i)*T(q, j) for gcd(p, q)=1.

T(n,k) = Sum_{d|k} mu(d/k) A316586(n,d).

A316537 Number of cyclic subgroups of the group SL(2, Z(n)), counting conjugates as distinct.

Original entry on oeis.org

1, 5, 13, 28, 49, 73, 116, 176, 202, 265, 378, 464, 550, 636, 842, 936, 1041, 1183, 1486, 1712, 2082, 2055, 2120, 3088, 2114, 3023, 2503, 4200, 4238, 4862, 4902, 4648, 6564, 5749, 7434, 7688, 6331, 8190, 9880, 11344, 10172, 12066, 9378, 13224, 14168, 11612

Offset: 1

Andrew Howroyd, Jul 06 2018

nonn

			Case n=2: generators of the 5 cyclic groups are:
  [ 1 0 ]   [0 1]   [1 0]   [1 1]   [0 1]
  [ 0 1 ]   [1 0]   [1 1]   [0 1]   [1 1]

Andrew Howroyd, Table of n, a(n) for n = 1..100

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

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

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

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

cp's OEIS Frontend

A316564 Triangle read by rows: T(n,k) is the number of elements of the group SL(2, Z(n)) with order k, 1 <= k <= A316563(n).

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