A316537 -id:A316537 - 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).

A316563 Maximum order of an element in the special linear group SL(2, Z(n)).

Original entry on oeis.org

1, 3, 6, 6, 10, 12, 14, 8, 18, 30, 22, 12, 26, 42, 30, 16, 34, 18, 38, 30, 42, 66, 46, 24, 50, 78, 54, 42, 58, 60, 62, 32, 66, 102, 70, 36, 74, 114, 78, 40, 82, 84, 86, 66, 90, 138, 94, 48, 98, 150, 102, 78, 106, 54, 110, 56, 114, 174, 118, 60, 122, 186, 126

Offset: 1

Andrew Howroyd, Jul 06 2018

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..100

Row lengths of A316564.

Cf. A000056, A316537, A316565.

Concatenation([1], List([2..15], n->Maximum(List(SL(2, Integers mod n), Order))));

MatOrder(M)={my(id=matid(#M), k=1, N=M); while(N<>id, k++;N=N*M); k}
a(n)={my(m=0); 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, m=max(m, MatOrder(M))))))); m}

A316553 Number of elements of order 2 in the group SL(2, Z(n)).

Original entry on oeis.org

0, 3, 1, 7, 1, 7, 1, 15, 1, 7, 1, 15, 1, 7, 3, 15, 1, 7, 1, 15, 3, 7, 1, 31, 1, 7, 1, 15, 1, 15, 1, 15, 3, 7, 3, 15, 1, 7, 3, 31, 1, 15, 1, 15, 3, 7, 1, 31, 1, 7, 3, 15, 1, 7, 3, 31, 3, 7, 1, 31, 1, 7, 3, 15, 3, 15, 1, 15, 3, 15, 1, 31, 1, 7, 3, 15, 3, 15, 1

Offset: 1

Andrew Howroyd, Jul 06 2018

nonn

Equivalently, the number of cyclic subgroups of the group SL(2, Z(n)) having order 2, counting conjugates as distinct.

			Case n=2: the three 2 X 2 matrices on Z(2) having determinant 1 and order 2 are:
  [ 0 1 ]   [ 1 0 ]   [ 1 1 ]
  [ 1 0 ]   [ 1 1 ]   [ 0 1 ]

Antti Karttunen, Table of n, a(n) for n = 1..1023

Column 2 of A316564.
Cf. A174275, A316537.
Cf. A061345.

Concatenation([0], List([2..10], n->Sum(Filtered( ConjugacyClassesSubgroups( SL(2, Integers mod n)), x->Order( Representative(x))=2 and IsCyclic( Representative(x))), Size)));

a(n)={my(id=matid(2)); 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, M^2==id))))) - 1}

memoA316553 = Map(); \\ Only values at 2^k are actually collected here.
A316553slow_memoized(n) = if(1==n, 0, if((n%2)&&isprimepower(n), 1, my(id=matid(2), v); if(mapisdefined(memoA316553,n,&v), v, v = (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, M^2==id))))) - 1); mapput(memoA316553,n,v); (v))));
A316553(n) = if(1==n,0,my(f=factor(n)); -1 + prod(i=1,#f~,1+A316553slow_memoized(f[i,1]^f[i,2]))); \\ (Based on Robert Israel's multiplicativity rule) - Antti Karttunen, Dec 05 2021

Conjecture: a(n) = 2^(omega(n) + min(3, valuation(n, 2))) - 1.
From Robert Israel, Jun 15 2020: (Start)
Number of solutions mod n, other than t[1]=t[4]=1,t[2]=t[3]=0, of the equations t[2]*(t[1] + t[4])=0, t[3]*(t[1] + t[4])=0, t[1]^2 + t[2]*t[3] = 1, t[2]*t[3] + t[4]^2 = 1, t[1]*t[4] - t[2]*t[3] = 1.
If m and n are coprime, a(m*n) = a(m)*a(n)+a(m)+a(n) (i.e. a(n)+1 is multiplicative).
If n > 1 is in A061345, a(n)=1. (End)

A316560 Number of cyclic subgroups of the group GL(2, Z(n)), counting conjugates as distinct.

Original entry on oeis.org

1, 5, 28, 62, 176, 148, 610, 696, 1252, 920, 2296, 1972, 4874, 3523, 6040, 6320, 8136, 7348, 14984, 13568, 22124, 11920, 17396, 23952, 29846, 28172, 38044, 47656, 47282, 32908, 75036, 53520, 71768, 42312, 145852, 99892, 123524, 88456, 187036, 179200, 152290

Offset: 1

Andrew Howroyd, Jul 06 2018

nonn

Cf. A053651, A066947, A316537, A316559, A316565, A316566.

Concatenation([1], List([2..7], n->Sum( Filtered( ConjugacyClassesSubgroups( GL(2, Integers mod n)), x->IsCyclic( Representative(x))), Size)));

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(gcd(lift(matdet(M)), n)==1, 1/eulerphi(MatOrder(M)))))))}

a(n) = Sum_{k=1..A316565(n)} 1/phi(A316566(n,k)).

A316536 Number of abelian subgroups of the group SL(2, Z(n)), counting conjugates as distinct.

Original entry on oeis.org

1, 5, 13, 42, 49, 105, 116, 498, 254, 381, 378, 1402, 550, 929, 1383, 3110, 1041, 2530, 1486, 5386, 3512, 2952, 2120, 24770

Offset: 1

Andrew Howroyd, Jul 06 2018

Cf. A316537, A316559.

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

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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A316563 Maximum order of an element in the special linear group SL(2, Z(n)).

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GAP

PARI

A316553 Number of elements of order 2 in the group SL(2, Z(n)).

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GAP

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PARI

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A316560 Number of cyclic subgroups of the group GL(2, Z(n)), counting conjugates as distinct.

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GAP

PARI

Formula

A316536 Number of abelian subgroups of the group SL(2, Z(n)), counting conjugates as distinct.

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Programs

GAP