A000003 -id:A000003 - cp's OEIS frontend

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

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).

A079029 Least k such that the class number of quadratic order of discriminant D=-4k equals p, where p runs through the primes.

Original entry on oeis.org

5, 11, 47, 71, 167, 191, 383, 311, 647, 887, 719, 1487, 1151, 1847, 3023, 2711, 2399, 3863, 3719, 5471, 2999, 4391, 3911, 6311, 5519, 5879, 13799, 8231, 5711, 8039, 19463, 12671, 15287, 9239, 17783, 22727, 25847, 40039, 15671, 14159, 17519, 14759, 22271, 26591

Offset: 1

Benoit Cloitre, Feb 01 2003

nonn

Cf. A000003.

a(n) = {my(k=1, p=prime(n)); while(abs(p-qfbclassno(-4*k))>0,k++); k; }

More terms from Jinyuan Wang, Apr 03 2020

A079030 Least k such that the class number of quadratic order of discriminant D=-4k equals n.

Original entry on oeis.org

1, 5, 11, 14, 47, 26, 71, 41, 59, 74, 167, 89, 191, 101, 131, 146, 383, 236, 311, 194, 251, 269, 647, 299, 479, 314, 419, 341, 887, 461, 719, 446, 659, 614, 1031, 626, 1487, 1199, 1019, 689, 1151, 794, 1847, 854, 971, 941, 3023, 1106, 1511, 1109, 1091, 1256, 2711

Offset: 1

Benoit Cloitre, Feb 01 2003

nonn

Cf. A000003.

a(n)=if(n<0,0,k=1; while(abs(n-qfbclassno(-4*k))>0,k++); k)

A250219 Number of times that n appears in n-th OEIS sequence (A_n), or -1 if n appears infinitely many times.

Original entry on oeis.org

-1, -1, 8, 0, -1, 4, 0, 1, 0, 2, 0, 0, 0, 1, 0, 1, 0, 0

Offset: 1

Eric Chen, Dec 24 2014

Does a(19) equal -1? (See A000019.)

			From _Jianing Song_, Sep 05 2018: (Start)
a(3) = 8 since A000003(k) = 3 for k = 11, 19, 23, 27, 31, 43, 67 and 163.
a(6) = 4 since A000006(k) = 6 for k = 12, 13, 14 and 15.
a(10) = 2 since A000010(k) = 10 for k = 11 and 22. (End)

Index entries for sequences whose definition involves A_n (or An).

Cf. A000001, A000002, A000003, A000004, etc.

a(3) corrected by Jianing Song, Sep 05 2018

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

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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Mathematica

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A079029 Least k such that the class number of quadratic order of discriminant D=-4k equals p, where p runs through the primes.

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PARI

Extensions

A079030 Least k such that the class number of quadratic order of discriminant D=-4k equals n.

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PARI

A250219 Number of times that n appears in n-th OEIS sequence (A_n), or -1 if n appears infinitely many times.

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Extensions

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

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

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

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