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A161010
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a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 14.
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4
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1, 8191, 797161, 33550336, 305175781, 6529545751, 16148168401, 137422176256, 423644039001, 2499694822171, 3452271214393, 26745019396096, 25239592216021, 132269647372591, 243274230757741, 562881233944576, 619036127056621
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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^13 such that the quotient group Z^13 / 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_13(n)/J_1(n) where J_13 and J_1(n) = A000010(n) are Jordan functions. - R. J. Mathar, Jul 12 2011
Multiplicative with a(p^e) = p^(12e-12) * (p^13-1) / (p-1).
For squarefree n, a(n) = A000203(n^12). (End)
Sum_{k=1..n} a(k) ~ c * n^13, where c = (1/13) * Product_{p prime} (1 + (p^12-1)/((p-1)*p^13)) = 0.14949521105... .
Sum_{k>=1} 1/a(k) = zeta(12)*zeta(13) * Product_{p prime} (1 - 2/p^13 + 1/p^25) = 1.0001233729754... . (End)
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MAPLE
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f:= proc(n) local t; mul(t[1]^(12*t[2]-12)*(t[1]^13-1)/(t[1]-1), t = ifactors(n)[2]) end proc:
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MATHEMATICA
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b = 14; Table[Sum[MoebiusMu[n/d] d^(b - 1), {d, Divisors@ n}]/EulerPhi@ n, {n, 17}] (* Michael De Vlieger, Nov 27 2015 *)
f[p_, e_] := p^(12*e - 12) * (p^13-1) / (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^12, d, if(ispower(d, 13), moebius(sqrtnint(d, 13))*sigma(n^12/d), 0))) \\ Altug Alkan, Nov 26 2015
(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i, 1]^13 - 1)*f[i, 1]^(12*f[i, 2] - 12)/(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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