A316566 Triangle read by rows: T(n,k) is the number of elements of the group GL(2, Z(n)) with order k, 1 <= k <= A316565(n).

1, 1, 3, 2, 1, 13, 8, 6, 0, 8, 0, 12, 1, 27, 8, 36, 0, 24, 1, 31, 20, 152, 24, 20, 0, 40, 0, 24, 0, 40, 0, 0, 0, 0, 0, 0, 0, 48, 0, 0, 0, 80, 1, 55, 26, 24, 0, 98, 0, 48, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 24, 1, 57, 170, 42, 0, 618, 48, 84, 0, 0, 0, 84

Offset: 1

Author

Andrew Howroyd, Jul 06 2018

Keywords

• nonn
• tabf

Comments

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

Examples

			Triangle begins:
  1
  1, 3, 2
  1, 13, 8, 6, 0, 8, 0, 12
  1, 27, 8, 36, 0, 24
  1, 31, 20, 152, 24, 20, 0, 40, 0, 24, 0, 40, 0, 0, 0, 0, 0, 0, 0, 48, 0, 0, 0, 80
  1, 55, 26, 24, 0, 98, 0, 48, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 24
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..8660 (first 40 rows)

Crossrefs

Row sums are A000252.

Column 2 is A066947.

Cf. A316560, A316564, A316565, A316584.

Programs

PARI

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(gcd(lift(matdet(M)), n)==1, my(t=MatOrder(M)); while(#L
				

Formula

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) * A316584(n,k).