A054729 -id:A054729

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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).

+10
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

Gheorghe Coserea, Table of n, a(n) for n = 0..200010

János A. Csirik, Joseph L. Wetherell, Michael E. Zieve, On the genera of X_0(N), arXiv:math/0006096 [math.NT], 2000.

FORMULA

A001617(a(A001617(n))) = A001617(n) and a(A054729(n)) = -1 for all n>=1. - Gheorghe Coserea, May 22 2016

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];

seq[51] (* 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(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));

return(inv);

};

seq(51) \\ Gheorghe Coserea, May 21 2016

CROSSREFS

Cf. A001617, A054727, A054729.

KEYWORD

sign

AUTHOR

Janos A. Csirik, Apr 21 2000

STATUS

approved

A054730

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

+10
2

49267, 74135, 94091, 96463, 102727, 107643, 118639, 138483, 145125, 181703, 182675, 208523, 221943, 237387, 240735, 245263, 255783, 267765, 269627, 272583, 277943, 280647, 283887, 286815, 309663, 313447, 322435, 326355, 336675, 347823, 352719

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

There are 4329 odd integers in the sequence less than 10^7. - Gheorghe Coserea, May 23 2016

REFERENCES

J. A. Csirik, The genus of X_0(N) is not 150, preprint, 2000.

LINKS

Gheorghe Coserea, Table of n, a(n) for n = 1..4329

J. A. Csirik, M. Zieve, and J. Wetherell, On the genera of X0(N), arXiv:math/0006096 [math.NT], 2000.

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;

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));

select(x->(x%2==1), apply(x->(x-1), Vec(select(x->x==-1, inv, 1))));

};

scan(400*1000)

CROSSREFS

Cf. A054726, A054727, A054729.

KEYWORD

nonn

AUTHOR

Janos A. Csirik, Apr 21 2000

EXTENSIONS

More terms from Gheorghe Coserea, May 23 2016

Offset corrected by Gheorghe Coserea, May 23 2016

STATUS

approved

A273445

a(n) is the number of solutions of the equation n = A001617(k).

+10
2

15, 12, 8, 11, 7, 14, 4, 13, 7, 12, 4, 15, 4, 9, 6, 10, 5, 16, 2, 20, 3, 14, 7, 11, 2, 13, 5, 11, 3, 14, 3, 9, 6, 13, 3, 17, 3, 14, 4, 10, 4, 20, 3, 15, 3, 12, 1, 15, 2, 20, 4, 11, 3, 13, 3, 16, 3, 12, 3, 15, 3, 12, 5, 9, 4, 15, 2, 14, 5, 17, 3, 13

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

The zeros of the sequence are given by A054729. The first five zeros of the sequence have indexes 150, 180, 210, 286, 304.

LINKS

Gheorghe Coserea, Table of n, a(n) for n = 0..100001

J. A. Csirik, M. Zieve, and J. Wetherell, On the genera of X0(N), arXiv:math/0006096 [math.NT], 2000.

FORMULA

a(n) = card {k, n = A001617(k)}.

EXAMPLE

For n = 0 the a(0) = 15 solutions are:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18, 25 (A091401).

For n = 1 the a(1) = 12 solutions are:

11, 14, 15, 17, 19, 20, 21, 24, 27, 32, 36, 49 (A091403).

For n = 2 the a(2) = 8 solutions are:

22, 23, 26, 28, 29, 31, 37, 50 (A091404).

MATHEMATICA

(* b = A001617 *) nmax = 71;

b[n_] := b[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];

Clear[f];

f[m_] := f[m] = Module[{}, A001617 = Array[b, m]; a[n_] := Count[A001617, n]; Table[a[n], {n, 0, nmax}]];

f[m = nmax]; f[m = m + nmax];

While[Print["m = ", m]; f[m] != f[m - nmax], m = m + nmax];

A273445 = f[m] (* Jean-François Alcover, Dec 16 2018, using Michael Somos' code for 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, 0), bnd = 12*n + 18*sqrtint(n) + 100, g);

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

if (g <= n, a[g+1]++));

return(a);

};

seq(72)

CROSSREFS

Cf. A001617, A054729, A091401, A091403, A091404.

KEYWORD

nonn

AUTHOR

Gheorghe Coserea, May 22 2016

STATUS

approved

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).

+10
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);