A300915 - OEIS

%I #30 Dec 01 2022 11:01:37

%S 1,6,12,24,60,72,168,96,324,360,660,288,1092,1008,720,768,2448,1944,

%T 3420,1440,2016,3960,6072,1152,7500,6552,8748,4032,12180,4320,14880,

%U 6144,7920,14688,10080,7776,25308,20520,13104,5760

%N Order of the group PSL(2,Z_n).

%C The projective special linear group PSL(2,Z_n) is the quotient group of SL(2,Z_n) with its center.  The center of SL(2,Z_n) is the group of scalar matrices whose diagonal entry is x in Z_n such that x^2 = 1.  The elements of PSL(2,Z_n) are equivalence classes of 2 X 2 matrices with entries in Z_n where two matrices are equivalent if one is a scalar multiple of the other.

%H Andrew Howroyd, <a href="/A300915/b300915.txt">Table of n, a(n) for n = 1..1000</a>

%H Geoffrey Critzer, <a href="https://esirc.emporia.edu/handle/123456789/3595">Combinatorics of Vector Spaces over Finite Fields</a>, Master's thesis, Emporia State University, 2018.

%F a(n) = A000056(n)/A060594(n).

%F Multiplicative with a(2) = 6, a(2^2) = 24, a(2^e) = 3*2^(3*e-4) for e > 2, and a(p^e) = (p^2-1)*p^(3*e-2)/2 for p > 2. - _Amiram Eldar_, Dec 01 2022

%t n := 2; nn = 40; \[Gamma][n_, q_] := Product[q^n - q^i, {i, 0, n - 1}]; Prepend[ Table[Product[ FactorInteger[m][[All, 1]][[j]]^(n^2 (FactorInteger[m][[All, 2]][[j]] - 1)) \[Gamma][n,FactorInteger[m][[All, 1]][[j]]], {j, 1, PrimeNu[m]}], {m, 2, nn}]/Table[EulerPhi[m], {m, 2, nn}]/ Table[Count[Mod[Select[Range[m], GCD[#, m] == 1 &]^n, m], 1], {m, 2, nn}], 1]

%t f[p_, e_] := (p^2-1)*p^(3*e-2)/2; f[2, e_] := Switch[e, 1, 6, 2, 24, _, 3*2^(3*e-4)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 40] (* _Amiram Eldar_, Dec 01 2022 *)

%o (PARI) a(n) = {my(f=factor(n)); prod(i=1, #f~, my([p,e]=f[i,]); (p^2-1)*p^(3*e-2)/if(p==2, 2^min(2, e-1), 2))} \\ _Andrew Howroyd_, Aug 01 2018

%Y Cf. A000056, A060594.

%K nonn,mult

%O 1,2

%A _Geoffrey Critzer_, Mar 16 2018

%E Keyword:mult added by _Andrew Howroyd_, Aug 01 2018