A316564 -id:A316564 - cp's OEIS frontend

A000056 Order of the group SL(2,Z_n).

1, 6, 24, 48, 120, 144, 336, 384, 648, 720, 1320, 1152, 2184, 2016, 2880, 3072, 4896, 3888, 6840, 5760, 8064, 7920, 12144, 9216, 15000, 13104, 17496, 16128, 24360, 17280, 29760, 24576, 31680, 29376, 40320, 31104, 50616, 41040, 52416, 46080, 68880, 48384, 79464

Offset: 1

N. J. A. Sloane

The number of equivalence classes of matrices modulo n of integer matrices with determinant 1 modulo n. - Michael Somos, Mar 20 2004
24 | a(n) if n > 2. - Michael Somos, Nov 15 2011
A divisibility sequence, that is, a(n) divides a(n*m) for all positive integers n and m. - Michael Somos, Jan 01 2017
The group SL(2,Z_2) is isomorphic to the symmetric group S_3. - Bernard Schott, Mar 15 2020
a(n) = [SL_2(Z) : Gamma(n)], index of the principal congruence subgroup of the special linear group over integers. - Andrey Zabolotskiy, Feb 14 2025

			G.f. = x + 6*x^2 + 24*x^3 + 48*x^4 + 120*x^5 + 144*x^6 + 336*x^7 +384*x^8 + ...
a(2) = 6 because [0, 1; 1, 0], [0, 1; 1, 1], [1, 0; 0, 1], [1, 0; 1, 1], [1, 1; 0, 1], [1, 1; 1, 0] are the six matrices modulo 2 with determinant 1 modulo 2.

T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, 1990, page 46.
B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 75.

T. D. Noe, Table of n, a(n) for n = 1..1000
Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003.
Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003 [Cached copy, with permission (pdf only)]
Wikipedia, Congruence subgroup.
Index to divisibility sequences
Index entries for sequences related to groups

Cf. A001766.
Row n=2 of A316623.
Row sums of A316564.
Cf. A000252 (GL(2,Z_n)), A064767 (GL(3,Z_n)), A305186 (GL(4,Z_n)).
Cf. A011785 (SL(3,Z_n)), A011786 (SL(4,Z_n)).
Cf. A007434 ([SL_2(Z) : Gamma_1(n)]), A001615 ([SL_2(Z) : Gamma_0(n)]).

proc(n) local b,d: b := n^3: for d from 1 to n do if irem(n,d) = 0 and isprime(d) then b := b*(1-d^(-2)): fi: od: RETURN(b): end:

(* From Olivier Gérard, Aug 15 1997: (Start) *)
Table[ Fold[ If[ Mod[ n, #2 ]==0 && PrimeQ[ #2 ], #1*(1-1/#2^2), #1 ]&, n^3, Range[ n ] ], {n, 1, 35} ]
Table[ n^3 Times@@(1-1/Select[ Range[ 1, n ], (Mod[ n, #1 ]==0&&PrimeQ[ #1 ])& ]^2), {n, 1, 35} ]  (* End *)
a[ n_] := If[ n<1, 0, n Sum[ d^2 MoebiusMu[ n/d ], {d, Divisors @ n}]]; (* Michael Somos, Nov 15 2011 *)
Table[ n DirichletConvolve[ MoebiusMu[m], m^2, m, n], {n, 1, 35}] (* Li Han, Mar 15 2020 *)
a[n_] := #.RotateLeft[#] & @ Sort[Mod[ Outer[Times, Range[n], Range[n]], n] // Flatten // Tally][[;; , 2]]
Table[a[n], {n, 1, 35}] (* Li Han, Mar 15 2020 *)

{a(n) = if( n<1, 0, n * sumdiv(n, d, d^2 * moebius(n / d)))}; /* Michael Somos, Mar 05 2008 */

from math import prod
from sympy import factorint
def A000056(n): return prod((p+1)*(p-1)*p**(3*e-2) for p,e in factorint(n).items()) # Chai Wah Wu, Mar 04 2025

Multiplicative with a(p^e) = (p^2 - 1)*p^(3e-2). - David W. Wilson, Aug 01 2001
a(n) = A000252(n)/phi(n), where phi is Euler totient function (cf. A000010). - Vladeta Jovovic, Oct 30 2001
a(n) = n*Sum_{d|n} d^2*mu(n/d) = n*A007434(n) where A007434 is the Jordan function J_2(n). - Benoit Cloitre, May 03 2003
a(n) = A007434(n^2)/n. - Enrique Pérez Herrero, Sep 14 2010
a(n) = A007434(n^3)/n^3. - Enrique Pérez Herrero, Dec 19 2010
Dirichlet g.f. zeta(s-3)/zeta(s-1). - R. J. Mathar, Feb 27 2011
A046970(n) divides a(n). - R. J. Mathar, Mar 30 2011
Sum_{k=1..n} a(k) ~ n^4 / (4*Zeta(3)). - Vaclav Kotesovec, Jan 30 2019
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^2 / ((p-1)^2 * (p+1) * (p^2 + p + 1))) = 1.258448350408311046314826069717731136828991478925039589864338603650639811... - Vaclav Kotesovec, Sep 19 2020

More terms from Vaclav Kotesovec, Sep 19 2020

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

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).

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

Andrew Howroyd, Jul 06 2018

nonn

			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]

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

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

Concatenation([1], List([2..10], n->Sum( Filtered( ConjugacyClassesSubgroups( SL(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(matdet(M)==1, 1/eulerphi(MatOrder(M)))))))}

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

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)

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

A000056 Order of the group SL(2,Z_n).

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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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A316537 Number of cyclic subgroups of the group SL(2, Z(n)), counting conjugates as distinct.

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

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

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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.

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