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

0,1

Comments

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

Links

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.

Formula

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

Example

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).

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 *)

Prog

(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)

Crossrefs

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

Sequence in context: A296818 A097953 A200522 * A338069 A195533 A299315

Adjacent sequences: A273442 A273443 A273444 * A273446 A273447 A273448

Keyword

nonn

Author

Gheorghe Coserea, May 22 2016

Status

approved