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A000114
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Number of cusps of principal congruence subgroup GAMMA^{hat}(n).
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2
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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
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OFFSET
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2,1
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COMMENTS
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REFERENCES
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R. A. Rankin, Modular Forms and Functions, Cambridge 1977, p. 62.
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LINKS
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FORMULA
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a(n) = ((n^2)/2)*Product_{p | n, p prime} (1-1/p^2), for n>=3. - Michel Marcus, Oct 23 2019
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MAPLE
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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:
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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