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

%I A116563 #31 Aug 04 2025 22:22:41
%S A116563 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,
%T A116563 11,12,12,12,13,14,14,15,14,16,15,16,16,17,18,19,18
%N A116563 a(n) is the genus of the modular curve X_0(p) for p = prime(n).
%C A116563 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
%C A116563 The previous name was "Genus of Ono X0[p] points". - _Felix Fröhlich_, May 21 2021
%H A116563 Mia Boudreau, <a href="/A116563/b116563.txt">Table of n, a(n) for n = 1..9999</a>
%H A116563 Ken Ono and Scott Ahlgren, <a href="https://web.archive.org/web/20190304015304/http://www.mathcs.emory.edu/~ono/publications-cv/pdfs/070.pdf">Weierstrass points on X0(p) and supersingular j-invariants</a>, Mathematische Annalen 325 (2003), 355-368, DOI:<a href="https://doi.org/10.1007/s00208-002-0390-9">10.1007/s00208-002-0390-9</a>.
%F A116563 From _Felix Fröhlich_, May 21 2021: (Start)
%F A116563 a(n) = A001617(prime(n)).
%F A116563 Let p = prime(n). Then
%F A116563 a(n) = (p-13)/12 if p == 1 (mod 12)
%F A116563 a(n) = (p-5)/12 if p == 5 (mod 12)
%F A116563 a(n) = (p-7)/12 if p == 7 (mod 12)
%F A116563 a(n) = (p+1)/12 if p == 11 (mod 12). (End)
%e A116563 a(707) = 445 as floor((prime(707) + 1) / 12) = 445, as 707 == 11 (mod 12) which maps to 1.
%t A116563 g[n_] := (Prime[n] - 13)/12 /; Mod[Prime[n], 12] - 1 == 0
%t A116563 g[n_] := (Prime[n] - 5)/12 /; Mod[Prime[n], 12] - 5 == 0
%t A116563 g[n_] := (Prime[n] - 7)/12 /; Mod[Prime[n], 12] - 7 == 0
%t A116563 g[n_] := (Prime[n] + 1)/12 /; Mod[Prime[n], 12] - 11 == 0
%t A116563 Table[g[n], {n, 3, 50}]
%o A116563 (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
%o A116563 (Java)
%o A116563 int a(int n){
%o A116563  int p = prime(n);
%o A116563  return (p - switch(p % 12){
%o A116563   case 1 -> 13; case 2 -> 5; case 3 -> 7; default -> -1;}) / 12;}
%Y A116563 Cf. A001617, A116569, A000040.
%K A116563 nonn
%O A116563 1,9
%A A116563 _Roger L. Bagula_, Mar 17 2006
%E A116563 Offset corrected by _Michel Marcus_, Apr 06 2018
%E A116563 Edited by _Felix Fröhlich_, May 21 2021