A065430 Order of commutator subgroup of GL(2,Z_n) (invertible 2 X 2 matrices mod n: A000252).

1, 3, 24, 24, 120, 72, 336, 192, 648, 360, 1320, 576, 2184, 1008, 2880, 1536, 4896, 1944, 6840, 2880, 8064, 3960, 12144, 4608, 15000, 6552, 17496, 8064, 24360, 8640, 29760, 12288, 31680, 14688, 40320, 15552, 50616, 20520, 52416, 23040, 68880

Offset: 1

Author

Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Nov 16 2001

Keywords

• nonn
• mult

Comments

This sequence may be multiplicative. - Mitch Harris, Apr 19 2005

Multiplicative because A000056 is. - Max Alekseyev

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000056, A000252, A002117.

Programs

Mathematica

Table[n DivisorSum[n, #^2 MoebiusMu[n/#] &]/(1 + Boole[EvenQ@ n]), {n, 41}] (* Michael De Vlieger, Mar 17 2018, after Harvey P. Dale at A000056 *)
f[p_, e_] := (p^2 - 1)*p^(3*e-2); f[2, e_] := 3*2^(3*e-3); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 30 2022 *)

PARI

sl(n) = n * sumdiv(n, d, d^2 * moebius(n / d));
a(n) = if (n%2, sl(n), sl(n)/2); \\ Michel Marcus, Mar 16 2018

Formula

For odd n: a(n) = A000056(n) i.e. the commutator subgroup is SL(2, Z_n);

for even n: a(n) = A000056(n) / 2 (it has index 2 in SL(2, Z_n)).

From Amiram Eldar, Nov 30 2022: (Start)

Multiplicative with a(2^e) = 3*2^(3*e-3), and a(p^e) = (p^2-1)*p^(3*e-2) if p > 2.

Sum_{k=1..n} a(k) ~ c * n^4, where c = 11/(56*zeta(3)) = 0.1634103... . (End)

Extensions

More terms from Max Alekseyev, Jan 22 2010