OFFSET
0,1
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
LINKS
Gheorghe Coserea, Table of n, a(n) for n = 0..200010
J. A. Csirik, M. Zieve, and J. Wetherell, On the genera of X0(N), arXiv:math/0006096 [math.NT], 2000.
FORMULA
EXAMPLE
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.
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 *)
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));
};
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)
CROSSREFS
KEYWORD
sign
AUTHOR
Gheorghe Coserea, May 23 2016
STATUS
approved