cp's OEIS Frontend

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

%I A065430 #24 Nov 30 2022 10:01:41
%S A065430 1,3,24,24,120,72,336,192,648,360,1320,576,2184,1008,2880,1536,4896,
%T A065430 1944,6840,2880,8064,3960,12144,4608,15000,6552,17496,8064,24360,8640,
%U A065430 29760,12288,31680,14688,40320,15552,50616,20520,52416,23040,68880
%N A065430 Order of commutator subgroup of GL(2,Z_n) (invertible 2 X 2 matrices mod n: A000252).
%C A065430 This sequence may be multiplicative. - _Mitch Harris_, Apr 19 2005
%C A065430 Multiplicative because A000056 is. - _Max Alekseyev_
%H A065430 Michael De Vlieger, <a href="/A065430/b065430.txt">Table of n, a(n) for n = 1..10000</a>
%F A065430 For odd n: a(n) = A000056(n) i.e. the commutator subgroup is SL(2, Z_n);
%F A065430 for even n: a(n) = A000056(n) / 2 (it has index 2 in SL(2, Z_n)).
%F A065430 From _Amiram Eldar_, Nov 30 2022: (Start)
%F A065430 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 A065430 Sum_{k=1..n} a(k) ~ c * n^4, where c = 11/(56*zeta(3)) = 0.1634103... . (End)
%t A065430 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 A065430 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 A065430 (PARI) sl(n) = n * sumdiv(n, d, d^2 * moebius(n / d));
%o A065430 a(n) = if (n%2, sl(n), sl(n)/2); \\ _Michel Marcus_, Mar 16 2018
%Y A065430 Cf. A000056, A000252, A002117.
%K A065430 nonn,mult
%O A065430 1,2
%A A065430 Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Nov 16 2001
%E A065430 More terms from _Max Alekseyev_, Jan 22 2010