A000114 - OEIS

%I #25 Oct 23 2019 15:53:31

%S 3,4,6,12,12,24,24,36,36,60,48,84,72,96,96,144,108,180,144,192,180,

%T 264,192,300,252,324,288,420,288,480,384,480,432,576,432,684,540,672,

%U 576,840,576,924,720,864,792,1104,768,1176,900,1152,1008,1404,972,1440

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

%C A001766(n) = n*a(n). - _Michael Somos_, Jan 29 2004

%D R. A. Rankin, Modular Forms and Functions, Cambridge 1977, p. 62.

%H N. J. A. Sloane, <a href="/A000114/b000114.txt">Table of n, a(n) for n = 2..10000</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 mu(n)/n p. 3.

%H A. Papoulis, <a href="/A000108/a000108_8.pdf">A new method of inversion of the Laplace transform</a>, Quart. Appl. Math 14 (1957), 405-414. [Annotated scan of selected pages]

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

%p 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:

%t 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 *)

%o (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

%Y Cf. A001766.

%K nonn,easy

%O 2,1

%A _N. J. A. Sloane_