A300915 Order of the group PSL(2,Z_n).

1, 6, 12, 24, 60, 72, 168, 96, 324, 360, 660, 288, 1092, 1008, 720, 768, 2448, 1944, 3420, 1440, 2016, 3960, 6072, 1152, 7500, 6552, 8748, 4032, 12180, 4320, 14880, 6144, 7920, 14688, 10080, 7776, 25308, 20520, 13104, 5760

Offset: 1

Author

Geoffrey Critzer, Mar 16 2018

Keywords

• nonn
• mult

Comments

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.

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1000
• Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.

Crossrefs

Cf. A000056, A060594.

Programs

Mathematica

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]
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 *)

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

Formula

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

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

Extensions

Keyword:mult added by Andrew Howroyd, Aug 01 2018