cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Views

Author

Gheorghe Coserea, Oct 22 2016

Keywords

Links

Crossrefs

Cf. A276183.

Programs

  • PARI
    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

Views

Author

Gheorghe Coserea, Oct 22 2016

Keywords

Links

Crossrefs

Cf. A276183.

Programs

  • PARI
    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

Views

Author

Gheorghe Coserea, Oct 22 2016

Keywords

Links

Crossrefs

Cf. A276183.

Programs

  • PARI
    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))

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

Views

Author

Gheorghe Coserea, Oct 17 2016

Keywords

Links

Crossrefs

Cf. A000003, A001617.

Programs

  • PARI
    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)))

Formula

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

Views

Author

David Jao, Aug 23 2023

Keywords

Links

Crossrefs

Cf. A000003, A001617, A029937, A276183.

Programs

  • Mathematica
    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]

Formula

a(n) = (A029937(n) - A001617(n))/2 + A276183(n).
Showing 1-5 of 5 results.

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