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A300915 Order of the group PSL(2,Z_n). 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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.

FORMULA

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

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]

PROG

(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

CROSSREFS

Cf. A000056, A060594.

Sequence in context: A082505 A091629 A089529 * A001766 A110959 A303398

Adjacent sequences:  A300912 A300913 A300914 * A300916 A300917 A300918

KEYWORD

nonn,mult

AUTHOR

Geoffrey Critzer, Mar 16 2018

EXTENSIONS

Keyword:mult added by Andrew Howroyd, Aug 01 2018

STATUS

approved

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Last modified October 17 14:47 EDT 2019. Contains 328114 sequences. (Running on oeis4.)