A000114 Number of cusps of principal congruence subgroup GAMMA^{hat}(n).

3, 4, 6, 12, 12, 24, 24, 36, 36, 60, 48, 84, 72, 96, 96, 144, 108, 180, 144, 192, 180, 264, 192, 300, 252, 324, 288, 420, 288, 480, 384, 480, 432, 576, 432, 684, 540, 672, 576, 840, 576, 924, 720, 864, 792, 1104, 768, 1176, 900, 1152, 1008, 1404, 972, 1440

Offset: 2

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

• Robert A. Rankin, Modular Forms and Functions, Cambridge 1977, p. 62.

Links

• N. J. A. Sloane, Table of n, a(n) for n = 2..10000
• 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)/n, p. 3.
• Athanasios Papoulis, A new method of inversion of the Laplace transform, Quart. Appl. Math., Vol. 14, No. 4 (1957), 405-414; Annotated scan of selected pages.

Crossrefs

Cf. A001766, A002117, A007434.

Programs

Maple

A000114 := proc(n) local b,d: if n = 2 then RETURN(3); else b := n^2/2; 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); fi: end:

Mathematica

a[n_] := If[n == 2, 3, b = n^2/2; For[d = 1, d <= n, d++, If[Mod[n, d] == 0 && PrimeQ[d], b = b*(1-d^-2)]]; b]; Table[a[n], {n, 2, 50}] (* Jean-François Alcover, Feb 04 2016, adapted from Maple *)

PARI

a(n) = if (n==2, 3, my(f=factor(n)); prod(k=1, #f~, 1-1/f[k,1]^2)*n^2/2); \\ Michel Marcus, Oct 23 2019

Formula

A001766(n) = n*a(n). - Michael Somos, Jan 29 2004

a(n) = ((n^2)/2)*Product_{p | n, p prime} (1-1/p^2), for n>=3. - Michel Marcus, Oct 23 2019

From Amiram Eldar, Jun 01 2025: (Start)

a(n) = A007434(n)/2 for n >= 3.

Sum_{k=1..n} a(k) ~ n^3 / (6*zeta(3)). (End)