A054729 Numbers n such that genus of modular curve X_0(N) is never equal to n.

150, 180, 210, 286, 304, 312, 336, 338, 348, 350, 480, 536, 570, 598, 606, 620, 666, 678, 706, 730, 756, 780, 798, 850, 876, 896, 906, 916, 970, 1014, 1026, 1046, 1106, 1144, 1170, 1176, 1186, 1188, 1224, 1244, 1260, 1320, 1350, 1356, 1366

Offset

1,1

Comments

"Looking further in the list of integers not of the form g0(N), we do eventually find some odd values, the first one occurring at the 3885th position. There are four such up to 10^5 (out of 9035 total missed values), namely 49267, 74135, 94091, 96463." (see Csirik link) - Gheorghe Coserea, May 21 2016.

a(1534734) = 9999996. - Gheorghe Coserea, May 23 2016

Links

Gheorghe Coserea, Table of n, a(n) for n = 1..20155

J. A. Csirik, M. Zieve, and J. Wetherell, On the genera of X0(N), arXiv:math/0006096 [math.NT], 2000.

Mathematica

a1617[n_] := a1617[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];
seq[n_] := Module[{inv, bnd}, inv[_] = -1; bnd = 12 n + 18 Floor[Sqrt[n]] + 100; For[k = 1, k <= bnd, k++, g = a1617[k]; If[g <= n && inv[g+1] == -1, inv[g+1] = k]]; (Position[Array[inv, n+1], -1] // Flatten)-1];
seq[1000] (* Jean-François Alcover, Nov 20 2018, after Gheorghe Coserea and Michael Somos in A001617 *)

Prog

(PARI)
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;
scan(n) = {
  my(inv = vector(n+1, g, -1), bnd = 12*n + 18*sqrtint(n) + 100, g);
  for (k = 1, bnd, g = A001617(k);
       if (g <= n && inv[g+1] == -1, inv[g+1] = k));
  apply(x->(x-1), Vec(select(x->x==-1, inv, 1)))
};
scan(1367)  \\ Gheorghe Coserea, May 21 2016

Crossrefs

Cf. A054726, A054728, A054730.

Sequence in context: A215689 A104262 A048706 * A116185 A008889 A184075

Adjacent sequences: A054726 A054727 A054728 * A054730 A054731 A054732

Keyword

nonn

Author

Janos A. Csirik, Apr 21 2000

Status

approved