A001766 Index of (the image of) the modular group Gamma(n) in PSL_2(Z). (Formerly M4098 N1700)

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

1,2

Comments

Equivalently, the degree of the modular curve X(N) as a cover of the j-line.

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.

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

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:

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 *)

Prog

(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

Crossrefs

Cf. A000056, A000114, A002117.

Sequence in context: A091629 A089529 A300915 * A110959 A303398 A244743

Adjacent sequences: A001763 A001764 A001765 * A001767 A001768 A001769

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Olivier Gérard, Aug 15 1997

Definition corrected by Mira Bernstein, May 30 2006

Status

approved