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

1,3

Comments

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)).

Links

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

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) A316586(n,d).

Example

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;
  ...

Prog

(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(matdet(M)==1, my(t=MatOrder(M)); while(#L<t, listput(L, 0)); L[t]++ ))))); Vec(L)}
for(n=1, 9, print(row(n)));

Crossrefs

Column 2 is A316553.

Row sums are A000056.

Cf. A316537, A316566, A316586.

Sequence in context: A229345 A240235 A092742 * A214742 A204124 A316674

Adjacent sequences: A316561 A316562 A316563 * A316565 A316566 A316567

Keyword

nonn,tabf

Author

Andrew Howroyd, Jul 06 2018

Status

approved