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 1-4 of 4 results.

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

Views

Author

Janos A. Csirik, Apr 21 2000

Keywords

Comments

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

References

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

Links

Crossrefs

Cf. A001617, A054727, A054729.

Programs

  • Mathematica
    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 *)
  • PARI
    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

Formula

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

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

Original entry on oeis.org

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

Views

Author

Janos A. Csirik, Apr 21 2000

Keywords

Comments

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

References

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

Links

Crossrefs

Cf. A054726, A054727, A054729.

Programs

  • PARI
    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)

Extensions

More terms from Gheorghe Coserea, May 23 2016
Offset corrected by Gheorghe Coserea, May 23 2016

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

Original entry on oeis.org

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

Views

Author

Gheorghe Coserea, May 22 2016

Keywords

Comments

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

Examples

			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).

Links

Crossrefs

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

Programs

  • Mathematica
    (* 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 *)
  • PARI
    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)

Formula

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

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

Views

Author

Gheorghe Coserea, May 23 2016

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • PARI
    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)

Formula

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.
Showing 1-4 of 4 results.

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