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 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). 1
 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 (list; graph; refs; listen; history; text; internal format)
 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 Let S(n) = {k, n = A001617(k)}; if the level set S(n) is not empty then a(n) = max S(n) and A054728(n) = min S(n) and A273445(n) = card S(n), otherwise a(n) = A054728(n) = -1 and A273445(n) = 0. 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. 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. 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)); }; 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) CROSSREFS Cf. A001617, A054728, A054729, A091401, A091403, A091404, A273445. Sequence in context: A090093 A004936 A062058 * A198591 A069063 A064937 Adjacent sequences:  A273507 A273508 A273509 * A273511 A273512 A273513 KEYWORD sign AUTHOR Gheorghe Coserea, May 23 2016 STATUS approved

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Last modified June 14 15:06 EDT 2021. Contains 345025 sequences. (Running on oeis4.)