A273510 a(n) is the largest level N such that genus of modular curve X_0(N) is n (or -1 if no such curve exists).
25, 49, 50, 64, 81, 75, 121, 100, 169, 128, 127, 147, 157, 163, 181, 193, 199, 289, 229, 243, 239, 257, 361, 283, 293, 313, 343, 337, 349, 353, 373, 379, 397, 409, 421, 529, 439, 457, 463, 467, 487, 499, 509, 523, 541, 547, 557, 577, 625, 601, 613, 619, 631, 643, 661, 673, 677, 691, 841, 667, 733
Offset: 0
Keywords
Examples
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. For n = 2 we have 2 = A001617(k) when k is 22, 23, 26, 28, 29, 31, 37, 50 (A091404); the largest of this values is 50 therefore a(2) = 50. For n = 150 (= A054729(1)) we have 150 <> A001617(k) for all k therefore a(150) = -1.
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.
Programs
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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 *) -
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; 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)
Comments