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A000114 Number of cusps of principal congruence subgroup GAMMA^{hat}(n). 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
2,1
COMMENTS
A001766(n) = n*a(n). - Michael Somos, Jan 29 2004
REFERENCES
R. A. Rankin, Modular Forms and Functions, Cambridge 1977, p. 62.
LINKS
Ioannis Ivrissimtzis, David Singerman, James Strudwick, From Farey fractions to the Klein quartic and beyond, arXiv:1909.08568 [math.GR], 2019. See mu(n)/n p. 3.
A. Papoulis, A new method of inversion of the Laplace transform, Quart. Appl. Math 14 (1957), 405-414. [Annotated scan of selected pages]
FORMULA
a(n) = ((n^2)/2)*Product_{p | n, p prime} (1-1/p^2), for n>=3. - Michel Marcus, Oct 23 2019
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 *)
PROG
(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
CROSSREFS
Cf. A001766.
Sequence in context: A175894 A175029 A113728 * A310006 A294144 A136243
KEYWORD
nonn,easy
AUTHOR
STATUS
approved

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Last modified October 4 08:18 EDT 2023. Contains 365873 sequences. (Running on oeis4.)