A001766 Index of (the image of) the modular group Gamma(n) in PSL_2(Z).
1, 6, 12, 24, 60, 72, 168, 192, 324, 360, 660, 576, 1092, 1008, 1440, 1536, 2448, 1944, 3420, 2880, 4032, 3960, 6072, 4608, 7500, 6552, 8748, 8064, 12180, 8640, 14880, 12288, 15840, 14688, 20160, 15552, 25308, 20520, 26208, 23040, 34440, 24192, 39732, 31680
Offset: 1
References
- R. C. Gunning, Lectures on Modular Forms, Princeton Univ. Press, Princeton, NJ, 1962, p. 15.
- B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 76.
- 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 = 1..1000
- Ioannis Ivrissimtzis, David Singerman, and James Strudwick, From Farey fractions to the Klein quartic and beyond, arXiv:1909.08568 [math.GR], 2019. See mu(n), p. 3.
- Index entries for sequences related to modular groups.
Programs
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Maple
proc(n) local b,d: b := (n^3)/2: for d from 1 to n do if irem(n,d) = 0 and isprime(d) then b := b*(1-d^(-2)): fi: od: RETURN(b): end:
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Mathematica
Table[ (n^3)/If[ n>2, 2, 1 ] Times@@(1-1/Select[ Range[ n ], (Mod[ n, #1 ]==0&&PrimeQ[ #1 ])& ]^2), {n, 1, 45} ] (* Olivier Gérard, Aug 15 1997 *) -
PARI
a(n) = if (n==1, 1, if (n==2, 6, my(f=factor(n)); prod(k=1, #f~, 1-1/f[k,1]^2)*n^3/2)); \\ Michel Marcus, Oct 23 2019
Formula
a(n) = A000056(n) for n = 2 and (1/2)*A000056(n) for n > 2 (since -I is contained in Gamma(2) but not in Gamma(n) for n > 2).
a(n) = n * A000114(n). - Michael Somos, Jan 29 2004
a(n) = ((n^3)/2)*Product_{p | n, p prime} (1-1/p^2), for n>=3. - Michel Marcus, Oct 23 2019
Sum_{k=1..n} a(k) ~ n^4 / (8*zeta(3)). - Amiram Eldar, Jun 01 2025
Extensions
More terms from Olivier Gérard, Aug 15 1997
Definition corrected by Mira Bernstein, May 30 2006
Comments