A065430 - OEIS

%I #24 Nov 30 2022 10:01:41

%S 1,3,24,24,120,72,336,192,648,360,1320,576,2184,1008,2880,1536,4896,

%T 1944,6840,2880,8064,3960,12144,4608,15000,6552,17496,8064,24360,8640,

%U 29760,12288,31680,14688,40320,15552,50616,20520,52416,23040,68880

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

%C This sequence may be multiplicative. - _Mitch Harris_, Apr 19 2005

%C Multiplicative because A000056 is. - _Max Alekseyev_

%H Michael De Vlieger, <a href="/A065430/b065430.txt">Table of n, a(n) for n = 1..10000</a>

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

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

%F From _Amiram Eldar_, Nov 30 2022: (Start)

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

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

%t 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 *)

%t 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 *)

%o (PARI) sl(n) = n * sumdiv(n, d, d^2 * moebius(n / d));

%o a(n) = if (n%2, sl(n), sl(n)/2); \\ _Michel Marcus_, Mar 16 2018

%Y Cf. A000056, A000252, A002117.

%K nonn,mult

%O 1,2

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

%E More terms from _Max Alekseyev_, Jan 22 2010