A054729 Numbers n such that genus of modular curve X_0(N) is never equal to n.
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
Offset: 1
Keywords
Links
- Gheorghe Coserea, Table of n, a(n) for n = 1..20155
- 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_] := 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 *) -
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
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