A316586 -id:A316586 - 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).

A316584 Array read by antidiagonals: T(n,k) is the number of elements x in GL(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, 14, 1, 1, 4, 9, 28, 1, 1, 1, 20, 9, 32, 1, 1, 6, 1, 64, 21, 56, 1, 1, 1, 30, 1, 184, 27, 58, 1, 1, 4, 1, 60, 25, 80, 171, 176, 1, 1, 3, 32, 1, 72, 1, 100, 33, 110, 1, 1, 4, 9, 64, 1, 180, 1, 640, 297, 128, 1, 1, 1, 14, 9, 224, 1, 846, 1, 164, 63, 134, 1

Offset: 1

Andrew Howroyd, Jul 07 2018

All columns are multiplicative.

Some terms of this sequence may also be computed using a formula given by Kent Morrison (section 1.11 and 2.5 in the reference). See A053725 for a PARI implementation.

			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  14   9   20    1   30   1   32   9   14 ...
    4 | 1  28   9   64    1   60   1   64   9   28 ...
    5 | 1  32  21  184   25   72   1  224  21   80 ...
    6 | 1  56  27   80    1  180   1  128  27   56 ...
    7 | 1  58 171  100    1  846  49  184 171   58 ...
    8 | 1 176  33  640    1  432   1 1024  33  176 ...
    9 | 1 110 297  164    1 1566   1  272 729  110 ...
   10 | 1 128  63  736   25  432   1  896  63  320 ...
   11 | 1 134 111  244 1325  354   1  464 111 5950 ...
   12 | 1 392  81 1280    1 1800   1 2048  81  392 ...
   13 | 1 184 549 1096    1 2736 469 1408 549  184 ...
   14 | 1 232 513  400    1 5076  49  736 513  232 ...
   15 | 1 448 189 3680   25 2160   1 7168 189 1120 ...
   ...

Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

Column 2 is A066907.

Cf. A053725, A316566, A316586.

T(n,k) = Sum_{d|k} A316566(n, d).

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

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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