A116569 a(n) = (x^3 - x) / 6, where x is the genus of the modular curve X_0(p) for p = prime(n).

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 4, 4, 10, 10, 20, 10, 20, 35, 20, 35, 56, 56, 56, 84, 84, 120, 84, 120, 165, 220, 220, 220, 286, 286, 286, 364, 455, 455, 560, 455, 680, 560, 680, 680, 816, 969, 1140, 969

Offset: 1

Author

Roger L. Bagula, Mar 18 2006

Keywords

• nonn

Comments

From Mia Boudreau, Jul 29 2025: (Start)

Previously named "Ono prime weight function divided by 6.".

See A001617 and A116563 for definition of genus of modular curve for X_0(n). (End)

Examples

			a(415) = 2218636 = (A116563(415)^3 - A116563(415)) / 6.
		

Links

• Mia Boudreau, Table of n, a(n) for n = 1..10000
• Ken Ono and Scott Ahlgren, Weierstrass points on X0(p) and supersingular j-invariants, Mathematische Annalen 325, 2003, pp. 355-368.

Crossrefs

Cf. A116563, A001617, A000040.

Programs

Java

long a(int n){
 long p = prime(n);
 long k = (p - switch((int)(p % 12)){
  case 1 -> 13; case 2 -> 5; case 3 -> 7; default -> -1;}) / 12;
 return k * (k - 1) * (k + 1) / 6;} // Mia Boudreau, Jul 29 2025

Mathematica

g[1] = 1; g[2] = 1;
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]*(g[n]^2 - 1)/6, {n, 1, 50}]

PARI

a(n) = {if (n < 3, g = 1, p = prime(n); m = p % 12; g = if (m==1, (p-13)/12, if (m==5, (p-5)/12, if (m==7, (p-7)/12, if (m==11, (p+1)/12))))); g*(g^2-1)/6;} \\ Michel Marcus, Apr 06 2018

Formula

a(n) = (A116563(n)^3 - A116563(n)) / 6. - Mia Boudreau, Jul 29 2025

Extensions

Offset corrected by Michel Marcus, Apr 06 2018