A001767 Genus of modular group Gamma(n) = genus of modular curve Chi(n). (Formerly M2459 N0976)

0, 0, 0, 0, 1, 3, 5, 10, 13, 26, 25, 50, 49, 73, 81, 133, 109, 196, 169, 241, 241, 375, 289, 476, 421, 568, 529, 806, 577, 1001, 833, 1081, 1009, 1393, 1081, 1768, 1441, 1849, 1633, 2451, 1729, 2850, 2281, 2809, 2641, 3773, 2689, 4215, 3301, 4321, 3865, 5500

Offset

2,6

Comments

In Klein and Fricke, the level n is called Stufenzahlen, the congruence group is denoted by Gamma_{n} and the genus is called Geschlecht and denoted by p. - Michael Somos, Nov 08 2014

References

R. C. Gunning, Lectures on Modular Forms. Princeton Univ. Press, Princeton, NJ, 1962, p. 15.

B. Iversen, Hyperbolic Geometry, Cambridge Univ. Press, 1992, see p. 238.

F. Klein and R. Fricke, Vorlesungen ueber die theorie der elliptischen modulfunctionen, Teubner, Leipzig, 1890, Vol. 1, see p. 398.

Russian Encyclopedia of Mathematics, Vol. 3, page 931.

B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 94.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

T. D. Noe, Table of n, a(n) for n = 2..1000

Ioannis Ivrissimtzis, David Singerman, James Strudwick, From Farey fractions to the Klein quartic and beyond, arXiv:1909.08568 [math.GR], 2019. See g(n) p. 3.

Index entries for sequences related to modular groups

Formula

a(n) = 1 + (n-6)*A000010(n)*A001615(n)/24, for n > 2. - Gheorghe Coserea, Oct 23 2016

Example

G.f. = x^6 + 3*x^7 + 5*x^8 + 10*x^9 + 13*x^10 + 26*x^11 + 25*x^12 + ...

Mathematica

Join[{0}, Table[1 + n^2 (n - 6)/24 Product[If[Mod[n, Prime[p]] == 0, 1 - 1/Prime[p]^2, 1], {p, PrimePi[n]}], {n, 3, 100}]] (* T. D. Noe, Aug 10 2012 *)
a[ n_] := If[ n < 3, 0, 1 + n^2 (n - 6)/24 Product[ If[ PrimeQ[p] && Divisible[n, p], 1 - 1/p^2, 1], {p, 2, n}]]; (* Michael Somos, Nov 08 2014 *)

Prog

(PARI) {a(n) = if(n<3, 0, 1 + n^2 * (n-6) / 24 * prod(p=2, n, if( isprime(p) && (n%p==0), 1 - 1/p^2, 1)))}; /* Michael Somos, May 19 2004 */
(PARI)
a(n) = {
  if (n < 6, return(0));
  my(f = factor(n), fsz = matsize(f)[1],
     g = prod(k=1, fsz, f[k, 1]),
     h = prod(k=1, fsz, sqr(f[k, 1]) - 1));
  return(1 + (n-6)*sqr(n\g)*h\24);
};
vector(52, n, a(n+1))  \\ Gheorghe Coserea, Oct 23 2016

Crossrefs

Sequence in context: A160792 A308759 A137395 * A360956 A273160 A285138

Adjacent sequences: A001764 A001765 A001766 * A001768 A001769 A001770

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved