A365138 Genus of the quotient of the modular curve X_1(n) by the Fricke involution.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 2, 1, 3, 1, 2, 3, 5, 2, 6, 4, 6, 5, 10, 4, 12, 8, 10, 10, 11, 8, 20, 13, 15, 12, 24, 12, 28, 17, 20, 22, 33, 18, 34, 23, 31, 27, 45, 25, 39, 29, 42, 39, 56, 28, 62, 44, 47, 46, 59, 39, 77, 51, 65, 48, 85, 48, 93, 66, 71, 67, 89, 60, 109

Offset

1,17

Links

Table of n, a(n) for n=1..79.

C. H. Kim and J. K. Koo, Estimation of Genus for Certain Arithmetic Groups, Communications in Algebra, 32:7 (2004), 2479-2495.

Formula

a(n) = (A029937(n) - A001617(n))/2 + A276183(n).

Mathematica

A000003[n_] :=
 Length[Select[
   Flatten[#, 1] &@
    Table[{i, j, (j^2 + 4 n)/(4 i)}, {i, Sqrt[4 n/3]}, {j, 1 - i, i}],
    Mod[#3, 1] == 0 && #3 >= # &&
       GCD[##] == 1 && ! (# == #3 && #2 < 0) & @@ # &]];
A001617[n_] :=
  If[n < 1, 0,
   1 + Sum[MoebiusMu[d]^2 n/d/12 - EulerPhi[GCD[d, n/d]]/2, {d,
      Divisors@n}] -
    Count[(#^2 - # + 1)/n & /@ Range[n], _?IntegerQ]/3 -
    Count[(#^2 + 1)/n & /@ Range[n], _?IntegerQ]/4];
A029937[n_] =
  If[n < 5, 0,
   1 + Sum[d^2*MoebiusMu[n/d]/24 - EulerPhi[d]*EulerPhi[n/d]/4, {d,
      Divisors[n]}]];
A276183[n_] :=
 If[0 <= n <= 4,
  0, (A001617[n] + 1)/2 -
   If[Mod[n, 8] == 3, 4, If[Mod[n, 8] == 7, 6, 3]]*A000003[n]/12];
A365138[n_] := (A029937[n] - A001617[n])/2 + A276183[n]

Crossrefs

Cf. A000003, A001617, A029937, A276183.

Sequence in context: A085392 A361201 A089384 * A228812 A341049 A144113

Adjacent sequences: A365135 A365136 A365137 * A365139 A365140 A365141

Keyword

nonn

Author

David Jao, Aug 23 2023

Status

approved