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A365138
Genus of the quotient of the modular curve X_1(n) by the Fricke involution.
1
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
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
KEYWORD
nonn
AUTHOR
David Jao, Aug 23 2023
STATUS
approved