cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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).

Original entry on oeis.org

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

Views

Author

Andrew Howroyd, Jul 06 2018

Keywords

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

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).

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

Views

Author

Andrew Howroyd, Jul 06 2018

Keywords

Examples

			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]

Links

Crossrefs

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

Programs

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

Formula

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

A316559 Number of abelian subgroups of the group GL(2, Z(n)), counting conjugates as distinct.

Original entry on oeis.org

1, 5, 34, 147, 251, 309, 890, 10683, 2440, 2487, 3088
Offset: 1

Views

Author

Andrew Howroyd, Jul 06 2018

Keywords

Crossrefs

Cf. A316536, A316560.

Programs

  • GAP
    Concatenation([1], List([2..7], n->Sum( Filtered( ConjugacyClassesSubgroups( GL(2, Integers mod n)), x->IsAbelian( Representative(x))), Size)));
Showing 1-3 of 3 results.

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