A316584 -id:A316584 - cp's OEIS frontend

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

Andrew Howroyd, Jul 06 2018

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)).

			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
  ...

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

Row sums are A000252.
Column 2 is A066947.
Cf. A316560, A316564, A316565, A316584.

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

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).

A316586 Array read by antidiagonals: T(n,k) is the number of elements x in SL(2,Z_n) with x^k == I mod n where I is the identity matrix.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 3, 2, 1, 1, 4, 9, 8, 1, 1, 1, 8, 9, 2, 1, 1, 6, 1, 32, 21, 8, 1, 1, 1, 18, 1, 32, 27, 2, 1, 1, 4, 1, 24, 25, 32, 57, 16, 1, 1, 3, 8, 1, 42, 1, 44, 33, 2, 1, 1, 4, 9, 32, 1, 108, 1, 160, 99, 8, 1, 1, 1, 2, 9, 32, 1, 114, 1, 56, 63, 2, 1

Offset: 1

Andrew Howroyd, Jul 07 2018

All columns are multiplicative.

			Array begins:
================================================
  n\k | 1  2   3   4   5   6   7   8   9  10
------+-----------------------------------------
    1 | 1  1   1   1   1   1   1   1   1   1 ...
    2 | 1  4   3   4   1   6   1   4   3   4 ...
    3 | 1  2   9   8   1  18   1   8   9   2 ...
    4 | 1  8   9  32   1  24   1  32   9   8 ...
    5 | 1  2  21  32  25  42   1  32  21  50 ...
    6 | 1  8  27  32   1 108   1  32  27   8 ...
    7 | 1  2  57  44   1 114  49 128  57   2 ...
    8 | 1 16  33 160   1 144   1 256  33  16 ...
    9 | 1  2  99  56   1 198   1  56 243   2 ...
   10 | 1  8  63 128  25 252   1 128  63 200 ...
   11 | 1  2 111 112 265 222   1 112 111 530 ...
   12 | 1 16  81 256   1 432   1 256  81  16 ...
   13 | 1  2 183 184   1 366 469 184 183   2 ...
   14 | 1  8 171 176   1 684  49 512 171   8 ...
   15 | 1  4 189 256  25 756   1 256 189 100 ...
   ...

Cf. A316564, A316584.

T(n,k) = Sum_{d|k} A316564(n, d).

Conjecture: T(p,p) = p^2 for p prime.

cp's OEIS Frontend

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).

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