A070732 Size of largest conjugacy class in the group GL(2,Z_n).

1, 3, 12, 12, 30, 36, 56, 48, 108, 90, 132, 144, 182, 168, 360, 192, 306, 324, 380, 360, 672, 396, 552, 576, 750, 546, 972, 672, 870, 1080, 992, 768, 1584, 918, 1680, 1296, 1406, 1140, 2184, 1440, 1722, 2016, 1892, 1584, 3240, 1656, 2256, 2304, 2744, 2250

Offset: 1

Author

Sharon Sela (sharonsela(AT)hotmail.com), May 14 2002

Keywords

• nonn
• mult
• easy

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A062354.

Cf. A000010, A000056.

Cf. A000082, A099892, A098198, A330523.

Programs

Mathematica

f[n_] := Block[{a = 1, b = FactorInteger[n]}, While[ Length[b] > 0, a = a*(b[[1, 1]] + 1)*b[[1, 1]]^(2b[[1, 2]] - If[ OddQ[ b[[1, 1]]], 1, 2]); b = Drop[b, 1]]; a]; Table[ f[n], {n, 1, 55}]
Table[n*Sum[d^2 MoebiusMu[n/d], {d, Divisors[n]}]/EulerPhi[2*n], {n, 1, 100}] (* Vaclav Kotesovec, Feb 01 2019 *)
f[p_, e_] := (p + 1)*p^(2*e - If[p == 2, 2, 1]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 14 2020 *)

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]+1)*f[i,1]^(2*f[i,2] - if(f[i,1]==2,2,1)));} \\ Amiram Eldar, Nov 05 2022

Formula

Multiplicative with a(p^e) = (p+1)*p^(2e - k), k = 1 if p is odd, k = 2 if p is 2.

a(n) = A000056(n)/A000010(2*n). - Vladeta Jovovic, Dec 22 2003

From R. J. Mathar, Apr 14 2011: (Start)

Dirichlet g.f.: (2^s-1)*zeta(s-1)*zeta(s-2)/((2^s+2)*zeta(2s-2)).

Dirichlet convolution of A000082 with a signed variant of A099892. (End)

Sum_{k=1..n} a(k) ~ 7*n^3 / (2*Pi^2). - Vaclav Kotesovec, Feb 01 2019

Sum_{n>=1} 1/a(n) = (13/11) * zeta(2)^2 * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) = (13/11) * A098198 * A330523 = 1.7136743536... . - Amiram Eldar, Nov 05 2022

Extensions

Edited by Robert G. Wilson v, May 20 2002