A054729 - OEIS

%I #30 Nov 20 2018 11:41:25

%S 150,180,210,286,304,312,336,338,348,350,480,536,570,598,606,620,666,

%T 678,706,730,756,780,798,850,876,896,906,916,970,1014,1026,1046,1106,

%U 1144,1170,1176,1186,1188,1224,1244,1260,1320,1350,1356,1366

%N Numbers n such that genus of modular curve X_0(N) is never equal to n.

%C "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.

%C a(1534734) = 9999996. - _Gheorghe Coserea_, May 23 2016

%H Gheorghe Coserea, <a href="/A054729/b054729.txt">Table of n, a(n) for n = 1..20155</a>

%H J. A. Csirik, M. Zieve, and J. Wetherell, <a href="http://arXiv.org/abs/math/0006096">On the genera of X0(N)</a>, arXiv:math/0006096 [math.NT], 2000.

%t 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];

%t 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];

%t seq[1000] (* _Jean-François Alcover_, Nov 20 2018, after _Gheorghe Coserea_ and _Michael Somos_ in A001617 *)

%o (PARI)

%o A000089(n) = {

%o   if (n%4 == 0 || n%4 == 3, return(0));

%o   if (n%2 == 0, n \= 2);

%o   my(f = factor(n), fsz = matsize(f)[1]);

%o   prod(k = 1, fsz, if (f[k,1] % 4 == 3, 0, 2));

%o };

%o A000086(n) = {

%o   if (n%9 == 0 || n%3 == 2, return(0));

%o   if (n%3 == 0, n \= 3);

%o   my(f = factor(n), fsz = matsize(f)[1]);

%o   prod(k = 1, fsz, if (f[k,1] % 3 == 2, 0, 2));

%o };

%o A001615(n) = {

%o   my(f = factor(n), fsz = matsize(f)[1],

%o      g = prod(k=1, fsz, (f[k,1]+1)),

%o      h = prod(k=1, fsz, f[k,1]));

%o   return((n*g)\h);

%o };

%o A001616(n) = {

%o   my(f = factor(n), fsz = matsize(f)[1]);

%o   prod(k = 1, fsz, f[k,1]^(f[k,2]\2) + f[k,1]^((f[k,2]-1)\2));

%o };

%o A001617(n) = 1 + A001615(n)/12 - A000089(n)/4 - A000086(n)/3 - A001616(n)/2;

%o scan(n) = {

%o   my(inv = vector(n+1,g,-1), bnd = 12*n + 18*sqrtint(n) + 100, g);

%o   for (k = 1, bnd, g = A001617(k);

%o        if (g <= n && inv[g+1] == -1, inv[g+1] = k));

%o   apply(x->(x-1), Vec(select(x->x==-1, inv, 1)))

%o };

%o scan(1367)  \\ _Gheorghe Coserea_, May 21 2016

%Y Cf. A054726, A054728, A054730.

%K nonn

%O 1,1

%A Janos A. Csirik, Apr 21 2000