A001617 -id:A001617 - cp's OEIS frontend

A054730 Odd n such that genus of modular curve X_0(N) is never equal to n.

49267, 74135, 94091, 96463, 102727, 107643, 118639, 138483, 145125, 181703, 182675, 208523, 221943, 237387, 240735, 245263, 255783, 267765, 269627, 272583, 277943, 280647, 283887, 286815, 309663, 313447, 322435, 326355, 336675, 347823, 352719

Offset: 1

Janos A. Csirik, Apr 21 2000

nonn

There are 4329 odd integers in the sequence less than 10^7. - Gheorghe Coserea, May 23 2016

J. A. Csirik, The genus of X_0(N) is not 150, preprint, 2000.

Gheorghe Coserea, Table of n, a(n) for n = 1..4329
J. A. Csirik, M. Zieve, and J. Wetherell, On the genera of X0(N), arXiv:math/0006096 [math.NT], 2000.

Cf. A054726, A054727, A054729.

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));
  select(x->(x%2==1), apply(x->(x-1), Vec(select(x->x==-1, inv, 1))));
};
scan(400*1000)

More terms from Gheorghe Coserea, May 23 2016

Offset corrected by Gheorghe Coserea, May 23 2016

A111248 Dimension of the space of weight 2 modular forms for Gamma_0(n).

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 3, 3, 3, 2, 5, 1, 4, 4, 5, 2, 7, 2, 6, 4, 5, 3, 8, 5, 5, 6, 7, 3, 10, 3, 8, 6, 6, 6, 12, 3, 7, 6, 10, 4, 12, 4, 9, 10, 8, 5, 14, 8, 13, 8, 10, 5, 15, 8, 12, 8, 9, 6, 18, 5, 10, 12, 14, 8, 16, 6, 12, 10, 16, 7, 20, 6, 11, 16, 13, 10, 18, 7, 18, 15, 12, 8, 22, 10, 13, 12, 16, 8, 26

Offset: 1

Steven Finch, Oct 31 2005

nonn

Equivalently, this is the dimension of the space of level n, weight 2 modular forms. - Steven Finch, Apr 03 2009

Klaus Brockhaus, Table of n, a(n) for n = 1..10000
S. R. Finch, Modular forms on SL_2(Z), December 28, 2005. [Cached copy, with permission of the author]

Cf. A001616, A001617. [Steven Finch, Apr 03 2009]

Cf. A006571.

[ Dimension(ModularForms(Gamma0(n), 2)) : n in [1..100] ]; // Klaus Brockhaus, Mar 10 2011

a(n) = A001617(n) + A001616(n) - 1. - Steven Finch, Apr 03 2009

More terms from Steven Finch, Apr 03 2009

A116569 a(n) = (x^3 - x) / 6, where x is the genus of the modular curve X_0(p) for p = prime(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 4, 4, 10, 10, 20, 10, 20, 35, 20, 35, 56, 56, 56, 84, 84, 120, 84, 120, 165, 220, 220, 220, 286, 286, 286, 364, 455, 455, 560, 455, 680, 560, 680, 680, 816, 969, 1140, 969

Offset: 1

Roger L. Bagula, Mar 18 2006

nonn

From Mia Boudreau, Jul 29 2025: (Start)
Previously named "Ono prime weight function divided by 6.".
See A001617 and A116563 for definition of genus of modular curve for X_0(n). (End)

			a(415) = 2218636 = (A116563(415)^3 - A116563(415)) / 6.

Mia Boudreau, Table of n, a(n) for n = 1..10000
Ken Ono and Scott Ahlgren, Weierstrass points on X0(p) and supersingular j-invariants, Mathematische Annalen 325, 2003, pp. 355-368.

Cf. A116563, A001617, A000040.

long a(int n){
 long p = prime(n);
 long k = (p - switch((int)(p % 12)){
  case 1 -> 13; case 2 -> 5; case 3 -> 7; default -> -1;}) / 12;
 return k * (k - 1) * (k + 1) / 6;} // Mia Boudreau, Jul 29 2025

g[1] = 1; g[2] = 1;
g[n_] := (Prime[n] - 13)/12 /; Mod[Prime[n], 12] - 1 == 0;
g[n_] := (Prime[n] - 5)/12 /; Mod[Prime[n], 12] - 5 == 0;
g[n_] := (Prime[n] - 7)/12 /; Mod[Prime[n], 12] - 7 == 0;
g[n_] := (Prime[n] + 1)/12 /; Mod[Prime[n], 12] - 11 == 0;
Table[g[n]*(g[n]^2 - 1)/6, {n, 1, 50}]

a(n) = {if (n < 3, g = 1, p = prime(n); m = p % 12; g = if (m==1, (p-13)/12, if (m==5, (p-5)/12, if (m==7, (p-7)/12, if (m==11, (p+1)/12))))); g*(g^2-1)/6;} \\ Michel Marcus, Apr 06 2018

a(n) = (A116563(n)^3 - A116563(n)) / 6. - Mia Boudreau, Jul 29 2025

Offset corrected by Michel Marcus, Apr 06 2018

A273510 a(n) is the largest level N such that genus of modular curve X_0(N) is n (or -1 if no such curve exists).

Original entry on oeis.org

25, 49, 50, 64, 81, 75, 121, 100, 169, 128, 127, 147, 157, 163, 181, 193, 199, 289, 229, 243, 239, 257, 361, 283, 293, 313, 343, 337, 349, 353, 373, 379, 397, 409, 421, 529, 439, 457, 463, 467, 487, 499, 509, 523, 541, 547, 557, 577, 625, 601, 613, 619, 631, 643, 661, 673, 677, 691, 841, 667, 733

Offset: 0

Gheorghe Coserea, May 23 2016

sign

a(10^7) = 120000007 is the largest value in the first 1+10^7 terms of the sequence.

The exception occurs first at a(150) = -1. - Georg Fischer, Feb 15 2019

			For n = 0 we have 0 = A001617(k) when k is 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18, 25 (A091401); the largest of this values is 25 therefore a(0) = 25.
For n = 1 we have 1 = A001617(k) when k is 11, 14, 15, 17, 19, 20, 21, 24, 27, 32, 36, 49 (A091403); the largest of this values is 49 therefore a(1) = 49.
For n = 2 we have 2 = A001617(k) when k is 22, 23, 26, 28, 29, 31, 37, 50 (A091404); the largest of this values is 50 therefore a(2) = 50.
For n = 150 (= A054729(1)) we have 150 <> A001617(k) for all k therefore a(150) = -1.

Gheorghe Coserea, Table of n, a(n) for n = 0..200010
J. A. Csirik, M. Zieve, and J. Wetherell, On the genera of X0(N), arXiv:math/0006096 [math.NT], 2000.

Cf. A001617, A054728, A054729, A091401, A091403, A091404, A273445.

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[{a, bnd}, a = Table[-1, {n+1}]; bnd = 12n + 18 Floor[Sqrt[n] ] + 100; For[k = 1, k <= bnd, k++, g = a1617[k]; If[g <= n, a[[g+1]] = k]]; a];
seq[60] (* Jean-François Alcover, Nov 20 2018, after Gheorghe Coserea and Michael Somos in A001617 *)

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;
seq(n) = {
  my(a = vector(n+1, g, -1), bnd = 12*n + 18*sqrtint(n) + 100, g);
  for (k = 1, bnd, g = A001617(k); if (g <= n, a[g+1] = k));
  return(a);
};
seq(60)

Let S(n) = {k, n = A001617(k)}; if the level set S(n) is not empty then a(n) = max S(n) and A054728(n) = min S(n) and A273445(n) = card S(n), otherwise a(n) = A054728(n) = -1 and A273445(n) = 0.

A276181 Fricke's 37 cases for two-valued modular equations.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 29, 31, 32, 35, 36, 39, 41, 47, 49, 50, 59, 71

Offset: 1

Gheorghe Coserea, Oct 17 2016

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

Cf. A000003, A001617.

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;
A276183(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);
};
select(x->(x>1), Vec(select(x->x==0, vector(100, n, A276183(n)), 1)))

Numbers n>1 such that 0 = A276183(n).

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

Original entry on oeis.org

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

David Jao, Aug 23 2023

nonn

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

Cf. A000003, A001617, A029937, A276183.

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]

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

A276182 Numbers N such that the modular curve X_0(N) is hyperelliptic.

Original entry on oeis.org

22, 23, 26, 28, 29, 30, 31, 33, 35, 37, 39, 40, 41, 46, 47, 48, 50, 59, 71

Offset: 1

Gheorghe Coserea, Oct 17 2016

"The only case where the hyperelliptic involution is not defined by an element of SL(2, R) is N=37."
"For N = 40, 48 the hyperelliptic involution v is not of Atkin-Lehner type. The remaining sixteen values are listed in the table below, together with their genera and hyperelliptic involutions v." (see Ogg link)
n    N    g    v
  22   2    11
  23   2    23
  26   2    26
  28   2    7
  29   2    29
  30   3    15
  31   2    31
  33   3    11
  35   3    35
 39   3    39
 41   3    41
 46   5    23
 47   4    47
 50   2    50
 59   5    59
 71   6    71

J. S. Balakrishnan, B. Mazur, and N. Dogra, Ogg's torsion conjecture: fifty years later, Bull. Amer. Math. Soc., 62:2 (2025), 235-268.

Andrew P. Ogg, Hyperelliptic modular curves, Bulletin de la S. M. F., 102 (1974), p. 449-462.

Cf. A001617, A260990.

A276184 Numbers n such that A276183(n) = 1.

Original entry on oeis.org

22, 28, 30, 33, 34, 37, 38, 40, 43, 44, 45, 48, 51, 53, 54, 55, 56, 61, 63, 64, 65, 75, 79, 81, 83, 89, 95, 101, 119, 131

Offset: 1

Gheorghe Coserea, Oct 22 2016

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

Cf. A276183.

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;
A276183(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);
};
Vec(select(x->x==1, vector(5000, n, A276183(n)), 1))

A276185 Numbers n such that A276183(n) = 2.

Original entry on oeis.org

42, 46, 52, 57, 62, 67, 68, 69, 72, 73, 74, 77, 80, 87, 91, 98, 103, 107, 111, 121, 125, 143, 167, 191

Offset: 1

Gheorghe Coserea, Oct 22 2016

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

Cf. A276183.

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;
A276183(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);
};
Vec(select(x->x==2, vector(500, n, A276183(n)), 1))

A276186 Numbers n such that A276183(n) = 3.

Original entry on oeis.org

58, 60, 66, 76, 85, 86, 96, 97, 99, 100, 104, 109, 113, 127, 128, 139, 149, 151, 169, 179, 239

Offset: 1

Gheorghe Coserea, Oct 22 2016

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

Cf. A276183.

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;
A276183(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);
};
Vec(select(x->x==3, vector(500, n, A276183(n)), 1))

cp's OEIS Frontend

A054730 Odd n such that genus of modular curve X_0(N) is never equal to n.

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A111248 Dimension of the space of weight 2 modular forms for Gamma_0(n).

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A116569 a(n) = (x^3 - x) / 6, where x is the genus of the modular curve X_0(p) for p = prime(n).

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A273510 a(n) is the largest level N such that genus of modular curve X_0(N) is n (or -1 if no such curve exists).

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A276181 Fricke's 37 cases for two-valued modular equations.

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

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A276182 Numbers N such that the modular curve X_0(N) is hyperelliptic.

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A276184 Numbers n such that A276183(n) = 1.

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