|
|
A054728
|
|
a(n) is the smallest level N such that genus of modular curve X_0(N) is n (or -1 if no such curve exists).
|
|
4
|
|
|
1, 11, 22, 30, 38, 42, 58, 60, 74, 66, 86, 78, 106, 105, 118, 102, 134, 114, 223, 132, 166, 138, 188, 156, 202, 168, 214, 174, 236, 186, 359, 204, 262, 230, 278, 222, 298, 240, 314, 246, 326, 210, 346, 270, 358, 282, 557, 306, 394, 312, 412, 318
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
a(150) = -1, a(n) > 0 for 0<=n<=149.
a(9999988) = 119999861 is the largest value in the first 1+10^7 terms of the sequence. - Gheorghe Coserea, May 24 2016
|
|
REFERENCES
|
J. A. Csirik, The genus of X_0(N) is not 150, preprint, 2000.
|
|
LINKS
|
János A. Csirik, Joseph L. Wetherell, Michael E. Zieve, On the genera of X_0(N), arXiv:math/0006096 [math.NT], 2000.
|
|
FORMULA
|
|
|
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[{inv, bnd}, inv = Table[-1, {n+1}]; bnd = 12n + 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]]; inv];
|
|
PROG
|
(PARI)
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));
};
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));
};
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);
};
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(inv = vector(n+1, g, -1), bnd = 12*n + 18*sqrtint(n) + 100, g);
if (g <= n && inv[g+1] == -1, inv[g+1] = k));
return(inv);
};
|
|
CROSSREFS
|
|
|
KEYWORD
|
sign
|
|
AUTHOR
|
Janos A. Csirik, Apr 21 2000
|
|
STATUS
|
approved
|
|
|
|