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

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• mult

Comments

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

Examples

			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.
		

References

• 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.

Links

• 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

Crossrefs

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)]).

Programs

Maple

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:

Mathematica

(* 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 *)

PARI

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

Python

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

Formula

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

Extensions

More terms from Vaclav Kotesovec, Sep 19 2020