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A160972
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a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 13.
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4
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1, 4095, 265720, 8386560, 61035156, 1088123400, 2306881200, 17175674880, 47071500840, 249938963820, 313842837672, 2228476723200, 1941507093540, 9446678514000, 16218261652320, 35175782154240, 36413889826860
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OFFSET
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1,2
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COMMENTS
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a(n) is the number of lattices L in Z^12 such that the quotient group Z^12 / L is C_n. - Álvar Ibeas, Nov 26 2015
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LINKS
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FORMULA
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a(n) = J_12(n)/J_1(n) where J_12 and J_1(n) = A000010(n) are Jordan functions. - R. J. Mathar, Jul 12 2011
Multiplicative with a(p^e) = p^(11e-11) * (p^12-1) / (p-1).
For squarefree n, a(n) = A000203(n^11). (End)
Sum_{k=1..n} a(k) ~ c * n^12, where c = (1/12) * Product_{p prime} (1 + (p^11-1)/((p-1)*p^12)) = 0.1619398772... .
Sum_{k>=1} 1/a(k) = zeta(11)*zeta(12) * Product_{p prime} (1 - 2/p^12 + 1/p^23) = 1.0002481006668... . (End)
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MATHEMATICA
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b = 13; Table[Sum[MoebiusMu[n/d] d^(b - 1)/EulerPhi@ n, {d, Divisors@ n}], {n, 17}] (* Michael De Vlieger, Nov 27 2015 *)
f[p_, e_] := p^(11*e - 11) * (p^12-1) / (p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 20] (* Amiram Eldar, Nov 08 2022 *)
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PROG
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(PARI) vector(100, n, sumdiv(n^11, d, if(ispower(d, 12), moebius(sqrtnint(d, 12))*sigma(n^11/d), 0))) \\ Altug Alkan, Nov 26 2015
(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i, 1]^12 - 1)*f[i, 1]^(11*f[i, 2] - 11)/(f[i, 1] - 1)); } \\ Amiram Eldar, Nov 08 2022
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CROSSREFS
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KEYWORD
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nonn,mult
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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