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%I M2459 N0976 #38 Oct 23 2019 11:09:57
%S 0,0,0,0,1,3,5,10,13,26,25,50,49,73,81,133,109,196,169,241,241,375,
%T 289,476,421,568,529,806,577,1001,833,1081,1009,1393,1081,1768,1441,
%U 1849,1633,2451,1729,2850,2281,2809,2641,3773,2689,4215,3301,4321,3865,5500
%N Genus of modular group Gamma(n) = genus of modular curve Chi(n).
%C 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
%D R. C. Gunning, Lectures on Modular Forms. Princeton Univ. Press, Princeton, NJ, 1962, p. 15.
%D B. Iversen, Hyperbolic Geometry, Cambridge Univ. Press, 1992, see p. 238.
%D F. Klein and R. Fricke, Vorlesungen ueber die theorie der elliptischen modulfunctionen, Teubner, Leipzig, 1890, Vol. 1, see p. 398.
%D Russian Encyclopedia of Mathematics, Vol. 3, page 931.
%D B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 94.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A001767/b001767.txt">Table of n, a(n) for n = 2..1000</a>
%H Ioannis Ivrissimtzis, David Singerman, James Strudwick, <a href="https://arxiv.org/abs/1909.08568">From Farey fractions to the Klein quartic and beyond</a>, arXiv:1909.08568 [math.GR], 2019. See g(n) p. 3.
%H <a href="/index/Gre#groups_modular">Index entries for sequences related to modular groups</a>
%F a(n) = 1 + (n-6)*A000010(n)*A001615(n)/24, for n > 2. - _Gheorghe Coserea_, Oct 23 2016
%e G.f. = x^6 + 3*x^7 + 5*x^8 + 10*x^9 + 13*x^10 + 26*x^11 + 25*x^12 + ...
%t 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 *)
%t 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 *)
%o (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 */
%o (PARI)
%o a(n) = {
%o if (n < 6, return(0));
%o my(f = factor(n), fsz = matsize(f)[1],
%o g = prod(k=1, fsz, f[k,1]),
%o h = prod(k=1, fsz, sqr(f[k,1]) - 1));
%o return(1 + (n-6)*sqr(n\g)*h\24);
%o };
%o vector(52, n, a(n+1)) \\ _Gheorghe Coserea_, Oct 23 2016
%K nonn,easy
%O 2,6
%A _N. J. A. Sloane_
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