A054728 -id:A054728 - cp's OEIS frontend

A001617 Genus of modular group Gamma_0(n). Or, genus of modular curve X_0(n).

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 2, 2, 1, 0, 2, 1, 2, 2, 3, 2, 1, 3, 3, 3, 1, 2, 4, 3, 3, 3, 5, 3, 4, 3, 5, 4, 3, 1, 2, 5, 5, 4, 4, 5, 5, 5, 6, 5, 7, 4, 7, 5, 3, 5, 9, 5, 7, 7, 9, 6, 5, 5, 8, 5, 8, 7, 11, 6, 7, 4, 9, 7, 11, 7, 10, 9, 9, 7, 11, 7, 10, 9, 11, 9, 9, 7, 7, 9, 7, 8, 15

Offset: 1

N. J. A. Sloane

Also the dimension of the space of cusp forms of weight two and level n. - Gene Ward Smith, May 23 2006

			G.f. = x^11 + x^14 + x^15 + x^17 + x^19 + x^20 + x^21 + 2*x^22 + 2*x^23 + ...

B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 103.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Gheorghe Coserea, Table of n, a(n) for n = 1..50000 (first 1000 terms from N. J. A. Sloane)
J. A. Csirik, M. Zieve, and J. Wetherell, On the genera of X0(N), arXiv:math/0006096 [math.NT], 2000.
Harriet Fell, Morris Newman, and Edward Ordman, Tables of genera of groups of linear fractional transformations, J. Res. Nat. Bur. Standards Sect. B 67B 1963 61-68.
Steven R. Finch, Modular forms on SL_2(Z), December 28, 2005. [Cached copy, with permission of the author]
Ralf Hemmecke, Peter Paule, and Silviu Radu, Construction of Modular Function Bases for Gamma_0(121) related to p*(11*n + 6), (2019).
Nicolas Allen Smoot, Computer algebra with the fifth operation: applications of modular functions to partition congruences, Ph. D. Thesis, Johannes Kepler Univ., Linz (Austria 2022), 33.
Index entries for sequences related to modular groups

Cf. A001615, A000089, A000086, A001616, A054728, A091401, A091403, A091404.

a := func< n | n lt 1 select 0 else Dimension( CuspForms( Gamma0(n), 2))>; /* Michael Somos, May 08 2015 */

nu2 := proc (n) # number of elliptic points of order two (A000089) local i, s; if modp(n,4) = 0 then RETURN(0) fi; s := 1; for i in divisors(n) do if isprime(i) and i > 2 then s := s*(1+eval(legendre(-1,i))) fi od; s end:
nu3 := proc (n) # number of elliptic points of order three (A000086) local d, s; if modp(n,9) = 0 then RETURN(0) fi; s := 1; for d in divisors(n) do if isprime(d) then s := s*(1+eval(legendre(-3,d))) fi od; s end:
nupara := proc (n) # number of parabolic cusps (A001616) local b, d; b := 0; for d to n do if modp(n,d) = 0 then b := b+eval(phi(gcd(d,n/d))) fi od; b end:
A001615 := proc(n) local i,j; j := n; for i in divisors(n) do if isprime(i) then j := j*(1+1/i); fi; od; j; end;
genx := proc (n) # genus of X0(n) (A001617) #1+1/12*psi(n)-1/4*nu2(n)-1/3*nu3(n)-1/2*nupara(n) end: 1+1/12*A001615(n)-1/4*nu2(n)-1/3*nu3(n)-1/2*nupara(n) end: # Gene Ward Smith, May 23 2006

nu2[n_] := Module[{i, s}, If[Mod[n, 4] == 0, Return[0]]; s = 1; Do[ If[ PrimeQ[i] && i > 2, s = s*(1 + JacobiSymbol[-1, i])], {i, Divisors[n]}]; s];
nu3[n_] := Module[{d, s}, If[Mod[n, 9] == 0, Return[0]]; s = 1; Do[ If[ PrimeQ[d], s = s*(1 + JacobiSymbol[-3, d])], {d, Divisors[n]}]; s];
nupara[n_] := Module[{b, d}, b = 0; For[d = 1, d <= n, d++, If[ Mod[n, d] == 0, b = b + EulerPhi[ GCD[d, n/d]]]]; b];
A001615[n_] := Module[{i, j}, j = n; Do[ If[ PrimeQ[i], j = j*(1 + 1/i)], {i, Divisors[n]}]; j];
genx[n_] := 1 + (1/12)*A001615[n] - (1/4)*nu2[n] - (1/3)*nu3[n] - (1/2)*nupara[n];
A001617 = Table[ genx[n], {n, 1, 102}] (* Jean-François Alcover, Jan 04 2012, after Gene Ward Smith's Maple program *)
a[ 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]; (* Michael Somos, May 08 2015 *)

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));
};
a(n) = 1 + A001615(n)/12 - A000089(n)/4 - A000086(n)/3 - A001616(n)/2;
vector(102, n, a(n))  \\ Gheorghe Coserea, May 20 2016

a(n) = 1 + A001615(n)/12 - A000089(n)/4 - A000086(n)/3 - A001616(n)/2.
From Gheorghe Coserea, May 20 2016: (Start)
limsup a(n) / (n*log(log(n))) = exp(Euler)/(2*Pi^2), where Euler is A001620.
a(n) >= (n-5*sqrt(n)-8)/12, with equality iff n = p^2 for prime p=1 (mod 12) (see A068228).
a(n) < n * exp(Euler)/(2*Pi^2) * (log(log(n)) + 2/log(log(n))) for n>=3 (see Csirik link).
(End)

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

Original entry on oeis.org

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

Janos A. Csirik, Apr 21 2000

nonn

"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

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.

Cf. A054726, A054728, A054730.

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 *)

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

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.

cp's OEIS Frontend

A001617 Genus of modular group Gamma_0(n). Or, genus of modular curve X_0(n).

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