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

%I A273445 #24 Dec 16 2018 03:29:25
%S A273445 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,
%T A273445 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,
%U A273445 16,3,12,3,15,3,12,5,9,4,15,2,14,5,17,3,13
%N A273445 a(n) is the number of solutions of the equation n = A001617(k).
%C A273445 The zeros of the sequence are given by A054729. The first five zeros of the sequence have indexes 150, 180, 210, 286, 304.
%H A273445 Gheorghe Coserea, <a href="/A273445/b273445.txt">Table of n, a(n) for n = 0..100001</a>
%H A273445 J. A. Csirik, M. Zieve, and J. Wetherell, <a href="http://arXiv.org/abs/math/0006096">On the genera of X0(N)</a>, arXiv:math/0006096 [math.NT], 2000.
%F A273445 a(n) = card {k, n = A001617(k)}.
%e A273445 For n = 0 the a(0) = 15 solutions are:
%e A273445 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18, 25 (A091401).
%e A273445 For n = 1 the a(1) = 12 solutions are:
%e A273445 11, 14, 15, 17, 19, 20, 21, 24, 27, 32, 36, 49 (A091403).
%e A273445 For n = 2 the a(2) = 8 solutions are:
%e A273445 22, 23, 26, 28, 29, 31, 37, 50 (A091404).
%t A273445 (* b = A001617 *) nmax = 71;
%t A273445 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];
%t A273445 Clear[f];
%t A273445 f[m_] := f[m] = Module[{}, A001617 = Array[b, m]; a[n_] := Count[A001617, n]; Table[a[n], {n, 0, nmax}]];
%t A273445 f[m = nmax]; f[m = m + nmax];
%t A273445 While[Print["m = ", m]; f[m] != f[m - nmax], m = m + nmax];
%t A273445 A273445 = f[m] (* _Jean-François Alcover_, Dec 16 2018, using _Michael Somos_' code for A001617 *)
%o A273445 (PARI)
%o A273445 A000089(n) = {
%o A273445   if (n%4 == 0 || n%4 == 3, return(0));
%o A273445   if (n%2 == 0, n \= 2);
%o A273445   my(f = factor(n), fsz = matsize(f)[1]);
%o A273445   prod(k = 1, fsz, if (f[k,1] % 4 == 3, 0, 2));
%o A273445 };
%o A273445 A000086(n) = {
%o A273445   if (n%9 == 0 || n%3 == 2, return(0));
%o A273445   if (n%3 == 0, n \= 3);
%o A273445   my(f = factor(n), fsz = matsize(f)[1]);
%o A273445   prod(k = 1, fsz, if (f[k,1] % 3 == 2, 0, 2));
%o A273445 };
%o A273445 A001615(n) = {
%o A273445   my(f = factor(n), fsz = matsize(f)[1],
%o A273445      g = prod(k=1, fsz, (f[k,1]+1)),
%o A273445      h = prod(k=1, fsz, f[k,1]));
%o A273445   return((n*g)\h);
%o A273445 };
%o A273445 A001616(n) = {
%o A273445   my(f = factor(n), fsz = matsize(f)[1]);
%o A273445   prod(k = 1, fsz, f[k,1]^(f[k,2]\2) + f[k,1]^((f[k,2]-1)\2));
%o A273445 };
%o A273445 A001617(n) = 1 + A001615(n)/12 - A000089(n)/4 - A000086(n)/3 - A001616(n)/2;
%o A273445 seq(n) = {
%o A273445   my(a = vector(n+1,g,0), bnd = 12*n + 18*sqrtint(n) + 100, g);
%o A273445   for (k = 1, bnd, g = A001617(k);
%o A273445        if (g <= n, a[g+1]++));
%o A273445   return(a);
%o A273445 };
%o A273445 seq(72)
%Y A273445 Cf. A001617, A054729, A091401, A091403, A091404.
%K A273445 nonn
%O A273445 0,1
%A A273445 _Gheorghe Coserea_, May 22 2016