A273445 - OEIS

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

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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));