A001617 -id:A001617 - cp's OEIS frontend

A273445 a(n) is the number of solutions of the equation n = A001617(k).

15, 12, 8, 11, 7, 14, 4, 13, 7, 12, 4, 15, 4, 9, 6, 10, 5, 16, 2, 20, 3, 14, 7, 11, 2, 13, 5, 11, 3, 14, 3, 9, 6, 13, 3, 17, 3, 14, 4, 10, 4, 20, 3, 15, 3, 12, 1, 15, 2, 20, 4, 11, 3, 13, 3, 16, 3, 12, 3, 15, 3, 12, 5, 9, 4, 15, 2, 14, 5, 17, 3, 13

Offset: 0

Gheorghe Coserea, May 22 2016

nonn

The zeros of the sequence are given by A054729. The first five zeros of the sequence have indexes 150, 180, 210, 286, 304.

			For n = 0 the a(0) = 15 solutions are:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18, 25 (A091401).
For n = 1 the a(1) = 12 solutions are:
11, 14, 15, 17, 19, 20, 21, 24, 27, 32, 36, 49 (A091403).
For n = 2 the a(2) = 8 solutions are:
22, 23, 26, 28, 29, 31, 37, 50 (A091404).

Gheorghe Coserea, Table of n, a(n) for n = 0..100001
J. A. Csirik, M. Zieve, and J. Wetherell, On the genera of X0(N), arXiv:math/0006096 [math.NT], 2000.

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

(* b = A001617 *) nmax = 71;
b[n_] := b[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];
Clear[f];
f[m_] := f[m] = Module[{}, A001617 = Array[b, m]; a[n_] := Count[A001617, n]; Table[a[n], {n, 0, nmax}]];
f[m = nmax]; f[m = m + nmax];
While[Print["m = ", m]; f[m] != f[m - nmax], m = m + nmax];
A273445 = f[m] (* Jean-François Alcover, Dec 16 2018, using Michael Somos' code for 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,0), bnd = 12*n + 18*sqrtint(n) + 100, g);
  for (k = 1, bnd, g = A001617(k);
       if (g <= n, a[g+1]++));
  return(a);
};
seq(72)

a(n) = card {k, n = A001617(k)}.

A091401 Numbers n such that genus of group Gamma_0(n) is zero.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18, 25

Offset: 1

N. J. A. Sloane, Mar 02 2004

Equivalently, numbers n such that genus of modular curve X_0(n) is zero.

G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1971, see Prop. 1.40 and 1.43.

Miranda C. N. Cheng, John F. R. Duncan and Jeffrey A Harvey, Umbral moonshine and the Niemeier lattices, Research in the Mathematical Sciences, 2014, 1:3; See Eq. (22). - _N. J. A. Sloane_, Jun 19 2014
K. Harada, "Moonshine" of Finite Groups, European Math. Soc., 2010, p. 15.
Yang-Hui He, John McKay, Sporadic and Exceptional, arXiv:1505.06742 [math.AG], 2015.
Robert S. Maier, On Rationally Parametrized Modular Equations, arXiv:math/0611041 [math.NT], 2006.
K. Ono, The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-Series, CBMS Regional Conference Series in Mathematics, vol. 102, American Mathematical Society, Providence, RI, 2004. See p. 110.
B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 103.

Cf. A001617, A001615, A000089, A000086, A001616, A091403.
The table below is a consequence of Theorem 7.3 in Maier's paper.
N        EntryID        K       alpha
1
      A127776        4096    1
      A276018        729     1
      A002894        256     1
      A276019        125     4
      A093388        72      1
      A276021        49      9
      A081085        32      1
      A006077        27      1
     A276020        20      2
     A276022        12      1
     A276177        13      36
     A276178        8       1
     A276179        6       1
     A276180        5       4

Flatten@ Position[#, 0] &@ Table[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], {n, 120}] (* Michael De Vlieger, Dec 05 2016, after Michael Somos at A001617 *)

Numbers n such that A001617(n) = 0.

A029937 Genus of modular curve X_1(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 1, 1, 2, 5, 2, 7, 3, 5, 6, 12, 5, 12, 10, 13, 10, 22, 9, 26, 17, 21, 21, 25, 17, 40, 28, 33, 25, 51, 25, 57, 36, 41, 45, 70, 37, 69, 48, 65, 55, 92, 52, 81, 61, 85, 78, 117, 57, 126, 91, 97

Offset: 1

N. J. A. Sloane

Also the dimension of the space of cusp forms of weight two on Gamma1(n). [Steven Finch, Apr 03 2009]

F. Hirzebruch et al., Manifolds and Modular Forms, Vieweg, 2nd ed. 1994, p. 161.

T. D. Noe, Table of n, a(n) for n = 1..1000
S. R. Finch, Modular forms on SL_2(Z), December 28, 2005. [Cached copy, with permission of the author]
Chang Heon Kim, Ja Kyung Koo, On the genus of some modular curves of level N, Bull Austral. Math. Soc. 54 (1996) 291-297.
A. V. Sutherland, Torsion subgroups of elliptic curves over number fields, 2012. - From _N. J. A. Sloane_, Feb 03 2013

Cf. A001617, A029938. [Steven Finch, Apr 03 2009]

with(numtheory); A029937 := proc(n) local i,j; j := 1+(1/24)*phi(n)*A001615(n); for i in divisors(n) do j := j-(1/4)*phi(i)*phi(n/i) od; j; end;

a[n_ /; n<5] = 0; a[n_] := 1+Sum[d^2*MoebiusMu[n/d]/24 - EulerPhi[d]*EulerPhi[n/d]/4, {d, Divisors[n]}]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jan 13 2014 *)

A029935(n) = {
  my(f = factor(n), fsz = matsize(f)[1],
     g = prod(k=1, fsz, f[k,1]),
     h = prod(k=1, fsz, sqr(f[k,1]-1)*f[k,2] + sqr(f[k,1])-1));
  return(h*n\sqr(g));
};
a(n) = {
  if (n < 5, return(0));
  my(f = factor(n), fsz = matsize(f)[1],
     g = prod(k=1, fsz, f[k,1]),
     h = prod(k=1, fsz, sqr(f[k,1]) - 1));
  return(1 + sqr(n\g)*h/24 - A029935(n)/4);
};
vector(63, n, a(n))  \\ Gheorghe Coserea, Oct 23 2016

a(n) = 1 + A115000(n) - A029935(n)/4, n > 4. [Kim and Koo, Theorem 1]

A276183 Genus of the quotient of the modular curve X_0(n) by the Fricke involution.

Original entry on oeis.org

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

Offset: 1

Gheorghe Coserea, Oct 21 2016

nonn

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, 1988).

			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 + ...

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.
Harvey Cohn, A Numerical Survey of the Reduction of Modular Curve Genus by Fricke's Involutions, Number Theory (New York Seminar 1989-1990), p. 100.
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.
Andrew P. Ogg, Automorphismes de courbes modulaires, Séminaire Delange-Pisot-Poitou. Théorie des nombres, vol. 16, no. 1 (1974-1975), talk no. 7, p. 1.

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

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-François Alcover at A000003 *)
ClassList[n_?Negative] :=
Select[Flatten[#, 1] &@Table[
    {i, j, (j^2 - n)/(4 i)}, {i, Sqrt[-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];
a[n_] := If[0 <= n <= 4, 0, (A001617[n] + 1)/2 - If[Mod[n, 8] == 3, 4, If[Mod[n, 8] == 7, 6, 3]] Length[ClassList[-4 n]]/12] (* David Jao, Sep 07 2020 *)

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))

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.

New name from David Jao, Sep 07 2020

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

A091403 Numbers n such that genus of group Gamma_0(n) is 1.

Original entry on oeis.org

11, 14, 15, 17, 19, 20, 21, 24, 27, 32, 36, 49

Offset: 1

N. J. A. Sloane, Mar 02 2004

I assume it is known that there are no further terms? A reference for this would be nice.

Available conductors for modular elliptic curves genus 1. [From Artur Jasinski, Jun 24 2010]

B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 103.
G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1971, see Prop. 1.40 and 1.43.

Cf. A001617, A001615, A000089, A000086, A001616, A091401, A091404.

a89[n_] := a89[n] = Product[{p, e} = pe; Which[p < 3 && e == 1, 1, p == 2 && e > 1, 0, Mod[p, 4] == 1, 2, Mod[p, 4] == 3, 0, True, a89[p^e]], {pe, FactorInteger[n]}];
a86[n_] := a86[n] = Product[{p, e} = pe; Which[p == 1 || p == 3 && e == 1, 1, p == 3 && e > 1, 0, Mod[p, 3] == 1, 2, Mod[p, 3] == 2, 0, True, a86[p^e]], {pe, FactorInteger[n]}];
a1615[n_] := n Sum[MoebiusMu[d]^2/d, {d, Divisors[n]}];
a1616[n_] := Sum[EulerPhi[GCD[ d, n/d]], {d, Divisors[n]}];
a1617[n_] := 1 + a1615[n]/12 - a89[n]/4 - a86[n]/3 - a1616[n]/2;
Position[Array[a1617, 100], 1] // Flatten (* Jean-François Alcover, Oct 18 2018 *)

Numbers n such that A001617(n) = 1.

A054728 a(n) is the smallest 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

1, 11, 22, 30, 38, 42, 58, 60, 74, 66, 86, 78, 106, 105, 118, 102, 134, 114, 223, 132, 166, 138, 188, 156, 202, 168, 214, 174, 236, 186, 359, 204, 262, 230, 278, 222, 298, 240, 314, 246, 326, 210, 346, 270, 358, 282, 557, 306, 394, 312, 412, 318

Offset: 0

Janos A. Csirik, Apr 21 2000

sign

a(150) = -1, a(n) > 0 for 0<=n<=149.

a(9999988) = 119999861 is the largest value in the first 1+10^7 terms of the sequence. - Gheorghe Coserea, May 24 2016

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

Gheorghe Coserea, Table of n, a(n) for n = 0..200010
János A. Csirik, Joseph L. Wetherell, Michael E. Zieve, On the genera of X_0(N), arXiv:math/0006096 [math.NT], 2000.

Cf. A001617, A054727, A054729.

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 = Table[-1, {n+1}]; bnd = 12n + 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]]; inv];
seq[51] (* 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(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));
  return(inv);
};
seq(51)  \\ Gheorghe Coserea, May 21 2016

A001617(a(A001617(n))) = A001617(n) and a(A054729(n)) = -1 for all n>=1. - Gheorghe Coserea, May 22 2016

A091404 Numbers n such that genus of group Gamma_0(n) is 2.

Original entry on oeis.org

22, 23, 26, 28, 29, 31, 37, 50

Offset: 1

N. J. A. Sloane, Mar 02 2004

I assume it is known that there are no further terms? A reference for this would be nice.

B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 103.
G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1971, see Prop. 1.40 and 1.43.

Cf. A001617, A001615, A000089, A000086, A001616, A091401, A091403.

a89[n_] := a89[n] = Product[{p, e} = pe; Which[p < 3 && e == 1, 1, p == 2 && e > 1, 0, Mod[p, 4] == 1, 2, Mod[p, 4] == 3, 0, True, a89[p^e]], {pe, FactorInteger[n]}];
a86[n_] := a86[n] = Product[{p, e} = pe; Which[p == 1 || p == 3 && e == 1, 1, p == 3 && e > 1, 0, Mod[p, 3] == 1, 2, Mod[p, 3] == 2, 0, True, a86[p^e]], {pe, FactorInteger[n]}];
a1615[n_] := n Sum[MoebiusMu[d]^2/d, {d, Divisors[n]}];
a1616[n_] := Sum[EulerPhi[GCD[d, n/d]], {d, Divisors[n]}];
a1617[n_] := 1 + a1615[n]/12 - a89[n]/4 - a86[n]/3 - a1616[n]/2;
Position[Array[a1617, 100], 2] // Flatten (* Jean-François Alcover, Oct 19 2018 *)

Numbers n such that A001617(n) = 2.

A127788 Dimension of the space of newforms of weight 2 and level n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 2, 1, 0, 2, 1, 0, 2, 1, 2, 1, 1, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 1, 1, 1, 4, 1, 1, 2, 3, 1, 4, 2, 3, 2, 3, 2, 5, 0, 4, 3, 3, 1, 5, 3, 5, 2, 3, 1, 6, 1, 5, 4, 3, 1, 5, 1, 6, 2, 2, 3, 7, 2, 5, 4, 5, 3, 7, 3, 7, 2, 5, 3, 7, 2, 7, 3, 4, 1, 8, 3

Offset: 1

Steven Finch, Apr 04 2007

nonn

"Newform" is meant in the sense of Atkin-Lehner, that is, a primitive Hecke eigenform relative to the subgroup Gamma_0(n).

			a(p) = A001617(p) for any prime p.
G.f. = x^11 + x^14 + x^15 + x^17 + x^19 + x^20 + x^21 + 2*x^23 + x^24 + ...

H. Cohen, Number Theory. Vol. II. Analytic and Modern Tools. Springer, 2007, pp. 496-497.
Toshitsune Miyake, Modular Forms, Springer-Verlag, 1989. See Table A.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
E. Halberstadt and A. Kraus, Courbes de Fermat: résultats et problèmes, J. Reine Angew. Math. 548 (2002) 167-234. [_Steven Finch_, Mar 27 2009]
Xian-Jin Li, An arithmetic formula for certain coefficients of the Euler product of Hecke polynomials, arXiv:math/0403148 [math.NT], 2004; J. Number Theory 113 (2005) 175-200. See Formula (5.8).
G. Martin, Dimensions of the spaces of cusp forms and newforms on Gamma_0(N) and Gamma_1(N), J. Numb. Theory 112 (2005) 298-331. [_Steven Finch_, Mar 27 2009]

Cf. A001617, A116563.

seq( g0star(2,N),N=1..80); # using the source in A063195 - R. J. Mathar, Jul 15 2015

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]; a[n_ /; n < 10] = 0; a[n_] := a[n] =  A001617[n] - Sum[a[m]*DivisorSigma[0, n/m], {m, Divisors[n][[2 ;; -2]]}]; Table[a[n], {n, 1, 102}] (* Jean-François Alcover, Sep 07 2015, A001617 code due to Michael Somos *)

{a(n) = my(v = [1, 3, 4, 6], A, p, e); if( n<1, 0, A = factor(n); for( k=1, matsize(A)[1], [p, e] = A[k,]; v[1] *= if( e==1, p-1, e==2, p^2-p-1, p^(e-3) * (p+1) * (p-1)^2); v[2] *= if( p==2, (e==3) - (e<3), e==1, kronecker(-4, p) - 1, e==2, -kronecker(-4, p)); v[3] *= if( p==3, (e==3) - (e<3), e==1, kronecker(-3, p) - 1, e==2, -kronecker(-3, p)); v[4] *= if( e%2, 0, e==2, p-2, p^(e/2-2) * (p-1)^2)); moebius(n) + (v[1] - v[2] - v[3] - v[4]) / 12 )}; /* Michael Somos, Jun 06 2015 */

a(n) = A001617(n) - sum a(m)*d(n/m), where the summation is over all divisors 1 < m < n of n and d is the divisor function.

A116563 a(n) is the genus of the modular curve X_0(p) for p = prime(n).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 5, 4, 5, 6, 5, 6, 7, 7, 7, 8, 8, 9, 8, 9, 10, 11, 11, 11, 12, 12, 12, 13, 14, 14, 15, 14, 16, 15, 16, 16, 17, 18, 19, 18

Offset: 1

Roger L. Bagula, Mar 17 2006

nonn

Also the dimension of the space of cusp forms of weight two and level p, where p=5, 7, 11, 13, ... ranges over all primes exceeding 3. - Steven Finch, Apr 04 2007

The previous name was "Genus of Ono X0[p] points". - Felix Fröhlich, May 21 2021

			a(707) = 445 as floor((prime(707) + 1) / 12) = 445, as 707 == 11 (mod 12) which maps to 1.

Mia Boudreau, Table of n, a(n) for n = 1..9999
Ken Ono and Scott Ahlgren, Weierstrass points on X0(p) and supersingular j-invariants, Mathematische Annalen 325 (2003), 355-368, DOI:10.1007/s00208-002-0390-9.

Cf. A001617, A116569, A000040.

int a(int n){
 int p = prime(n);
 return (p - switch(p % 12){
  case 1 -> 13; case 2 -> 5; case 3 -> 7; default -> -1;}) / 12;}

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], {n, 3, 50}]

a(n) = {my(p = prime(n), m = p % 12); if (m==1, (p-13)/12, if (m==5, (p-5)/12, if (m==7, (p-7)/12, if (m==11, (p+1)/12))));} \\ Michel Marcus, Apr 06 2018

From Felix Fröhlich, May 21 2021: (Start)
a(n) = A001617(prime(n)).
Let p = prime(n). Then
a(n) = (p-13)/12 if p == 1 (mod 12)
a(n) = (p-5)/12 if p == 5 (mod 12)
a(n) = (p-7)/12 if p == 7 (mod 12)
a(n) = (p+1)/12 if p == 11 (mod 12). (End)

Offset corrected by Michel Marcus, Apr 06 2018

Edited by Felix Fröhlich, May 21 2021

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