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A276183 a(n) is the genus of quotient space H/Gamma_0*(n), where H is the upper half plane and Gamma_0*(n) = Gamma_0(n) + W Gamma_0(n) is the extension of Gamma_0(n) via the involution z <-> W(z) = -n/z (see Cohn link). 5
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 2, 1, 1, 1, 2, 0, 1, 0, 0, 1, 2, 1, 1, 1, 1, 2, 3, 0, 3, 1, 2, 1, 1, 1, 3, 2, 2, 2, 4, 0, 2, 2, 2, 1, 3, 2, 5, 1, 2, 1, 4, 1, 4, 3, 3, 2, 4, 1, 4, 2, 4, 4, 4, 1, 3, 3, 2, 3, 3, 1, 7 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,42

LINKS

Gheorghe Coserea, Table of n, a(n) for n = 1..54321

Harvey Cohn, Fricke's Two-Valued Modular Equations, Math. Comp. 51 (1988), 787-807.

Fell, Harriet; Newman, Morris; Ordman, Edward; Tables of genera of groups of linear fractional transformations, J. Res. Nat. Bur. Standards Sect. B 67B 1963 61-68.

FORMULA

a(n) = (1 + A001617(n))/2 - r * A000003(n)/12 for all n > 4, where r=4 for n=3 (mod 8), r=6 for n=7 (mod 8) and r=3 otherwise.

a(n) <> 4884 for all n.

EXAMPLE

G.f. = x^22 + x^28 + x^30 + x^33 + x^34 + x^37 + x^38 + x^40 + 2*x^42 + x^43 + x^44 + ...

MATHEMATICA

f[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];

g[n_] := Ceiling[k0 = k /. FindRoot[EllipticK[1 - k^2]/EllipticK[k^2] == Sqrt@ n, {k, 1/2, 10^-10, 1}, WorkingPrecision -> 600, MaxIterations -> 100]; Exponent[MinimalPolynomial[RootApproximant[k0^2, 24], x], x]/2];

r[n_] := If[MemberQ[{3, 7}, #], 3 + (# - 1)/2, 3] &@ Mod[n, 8]; a[n_] := If[n <= 4, 0, (1 + f@ n)/2 - r[n] g[n]/12]; Table[Print["a(", n, ") = ", an = a[n]]; an, {n, 102}] (* Michael De Vlieger, Oct 28 2016, after Michael Somos at A001617 and Jean-Fran├žois Alcover at A000003 *)

PROG

(PARI)

A000003(n) = qfbclassno(-4*n);

A000089(n) = {

  if (n%4 == 0 || n%4 == 3, return(0));

  if (n%2 == 0, n \= 2);

  my(f = factor(n), fsz = matsize(f)[1]);

  prod(k = 1, fsz, if (f[k, 1] % 4 == 3, 0, 2));

};

A000086(n) = {

  if (n%9 == 0 || n%3 == 2, return(0));

  if (n%3 == 0, n \= 3);

  my(f = factor(n), fsz = matsize(f)[1]);

  prod(k = 1, fsz, if (f[k, 1] % 3 == 2, 0, 2));

};

A001615(n) = {

  my(f = factor(n), fsz = matsize(f)[1],

     g = prod(k=1, fsz, (f[k, 1]+1)),

     h = prod(k=1, fsz, f[k, 1]));

  return((n*g)\h);

};

A001616(n) = {

  my(f = factor(n), fsz = matsize(f)[1]);

  prod(k = 1, fsz, f[k, 1]^(f[k, 2]\2) + f[k, 1]^((f[k, 2]-1)\2));

};

A001617(n) = 1 + A001615(n)/12 - A000089(n)/4 - A000086(n)/3 - A001616(n)/2;

a(n) = {

  my(r = if (n%8 == 3, 4, n%8 == 7, 6, 3));

  if (n < 5, 0, (1 + A001617(n))/2 - r * A000003(n)/12);

};

vector(102, n, a(n))

CROSSREFS

Cf. A000003, A000086, A000089, A001615, A001616, A001617, A276181.

Sequence in context: A143277 A292378 A320835 * A056563 A088231 A327954

Adjacent sequences:  A276180 A276181 A276182 * A276184 A276185 A276186

KEYWORD

nonn

AUTHOR

Gheorghe Coserea, Oct 21 2016

STATUS

approved

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Last modified April 7 01:03 EDT 2020. Contains 333291 sequences. (Running on oeis4.)