A116563 a(n) is the genus of the modular curve X_0(p) for p = prime(n).

0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 5, 4, 5, 6, 5, 6, 7, 7, 7, 8, 8, 9, 8, 9, 10, 11, 11, 11, 12, 12, 12, 13, 14, 14, 15, 14, 16, 15, 16, 16, 17, 18, 19, 18

Offset: 1

Author

Roger L. Bagula, Mar 17 2006

Keywords

• nonn

Comments

Also the dimension of the space of cusp forms of weight two and level p, where p=5, 7, 11, 13, ... ranges over all primes exceeding 3. - Steven Finch, Apr 04 2007

The previous name was "Genus of Ono X0[p] points". - Felix Fröhlich, May 21 2021

Examples

			a(707) = 445 as floor((prime(707) + 1) / 12) = 445, as 707 == 11 (mod 12) which maps to 1.
		

Links

• Mia Boudreau, Table of n, a(n) for n = 1..9999
• Ken Ono and Scott Ahlgren, Weierstrass points on X0(p) and supersingular j-invariants, Mathematische Annalen 325 (2003), 355-368, DOI:10.1007/s00208-002-0390-9.

Crossrefs

Cf. A001617, A116569, A000040.

Programs

Java

int a(int n){
 int p = prime(n);
 return (p - switch(p % 12){
  case 1 -> 13; case 2 -> 5; case 3 -> 7; default -> -1;}) / 12;}

Mathematica

g[n_] := (Prime[n] - 13)/12 /; Mod[Prime[n], 12] - 1 == 0
g[n_] := (Prime[n] - 5)/12 /; Mod[Prime[n], 12] - 5 == 0
g[n_] := (Prime[n] - 7)/12 /; Mod[Prime[n], 12] - 7 == 0
g[n_] := (Prime[n] + 1)/12 /; Mod[Prime[n], 12] - 11 == 0
Table[g[n], {n, 3, 50}]

PARI

a(n) = {my(p = prime(n), m = p % 12); if (m==1, (p-13)/12, if (m==5, (p-5)/12, if (m==7, (p-7)/12, if (m==11, (p+1)/12))));} \\ Michel Marcus, Apr 06 2018

Formula

From Felix Fröhlich, May 21 2021: (Start)

a(n) = A001617(prime(n)).

Let p = prime(n). Then

a(n) = (p-13)/12 if p == 1 (mod 12)

a(n) = (p-5)/12 if p == 5 (mod 12)

a(n) = (p-7)/12 if p == 7 (mod 12)

a(n) = (p+1)/12 if p == 11 (mod 12). (End)

Extensions

Offset corrected by Michel Marcus, Apr 06 2018

Edited by Felix Fröhlich, May 21 2021