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A054729 Numbers n such that genus of modular curve X_0(N) is never equal to n. 5
150, 180, 210, 286, 304, 312, 336, 338, 348, 350, 480, 536, 570, 598, 606, 620, 666, 678, 706, 730, 756, 780, 798, 850, 876, 896, 906, 916, 970, 1014, 1026, 1046, 1106, 1144, 1170, 1176, 1186, 1188, 1224, 1244, 1260, 1320, 1350, 1356, 1366 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
"Looking further in the list of integers not of the form g0(N), we do eventually find some odd values, the first one occurring at the 3885th position. There are four such up to 10^5 (out of 9035 total missed values), namely 49267, 74135, 94091, 96463." (see Csirik link) - Gheorghe Coserea, May 21 2016.
a(1534734) = 9999996. - Gheorghe Coserea, May 23 2016
LINKS
J. A. Csirik, M. Zieve, and J. Wetherell, On the genera of X0(N), arXiv:math/0006096 [math.NT], 2000.
MATHEMATICA
a1617[n_] := 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[_] = -1; bnd = 12 n + 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]]; (Position[Array[inv, n+1], -1] // Flatten)-1];
seq[1000] (* 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));
};
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));
apply(x->(x-1), Vec(select(x->x==-1, inv, 1)))
};
scan(1367) \\ Gheorghe Coserea, May 21 2016
CROSSREFS
Sequence in context: A215689 A104262 A048706 * A116185 A008889 A184075
KEYWORD
nonn
AUTHOR
Janos A. Csirik, Apr 21 2000
STATUS
approved

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Last modified March 19 01:34 EDT 2024. Contains 370952 sequences. (Running on oeis4.)