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

%I A116569 #28 Aug 04 2025 22:24:47
%S A116569 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,
%T A116569 120,84,120,165,220,220,220,286,286,286,364,455,455,560,455,680,560,
%U A116569 680,680,816,969,1140,969
%N A116569 a(n) = (x^3 - x) / 6, where x is the genus of the modular curve X_0(p) for p = prime(n).
%C A116569 From _Mia Boudreau_, Jul 29 2025: (Start)
%C A116569 Previously named "Ono prime weight function divided by 6.".
%C A116569 See A001617 and A116563 for definition of genus of modular curve for X_0(n). (End)
%H A116569 Mia Boudreau, <a href="/A116569/b116569.txt">Table of n, a(n) for n = 1..10000</a>
%H A116569 Ken Ono and Scott Ahlgren, <a href="https://uva.theopenscholar.com/files/ken-ono/files/070.pdf">Weierstrass points on X0(p) and supersingular j-invariants</a>, Mathematische Annalen 325, 2003, pp. 355-368.
%F A116569 a(n) = (A116563(n)^3 - A116563(n)) / 6. - _Mia Boudreau_, Jul 29 2025
%e A116569 a(415) = 2218636 = (A116563(415)^3 - A116563(415)) / 6.
%t A116569 g[1] = 1; g[2] = 1;
%t A116569 g[n_] := (Prime[n] - 13)/12 /; Mod[Prime[n], 12] - 1 == 0;
%t A116569 g[n_] := (Prime[n] - 5)/12 /; Mod[Prime[n], 12] - 5 == 0;
%t A116569 g[n_] := (Prime[n] - 7)/12 /; Mod[Prime[n], 12] - 7 == 0;
%t A116569 g[n_] := (Prime[n] + 1)/12 /; Mod[Prime[n], 12] - 11 == 0;
%t A116569 Table[g[n]*(g[n]^2 - 1)/6, {n, 1, 50}]
%o A116569 (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
%o A116569 (Java)
%o A116569 long a(int n){
%o A116569  long p = prime(n);
%o A116569  long k = (p - switch((int)(p % 12)){
%o A116569   case 1 -> 13; case 2 -> 5; case 3 -> 7; default -> -1;}) / 12;
%o A116569  return k * (k - 1) * (k + 1) / 6;} // _Mia Boudreau_, Jul 29 2025
%Y A116569 Cf. A116563, A001617, A000040.
%K A116569 nonn
%O A116569 1,13
%A A116569 _Roger L. Bagula_, Mar 18 2006
%E A116569 Offset corrected by _Michel Marcus_, Apr 06 2018