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A161167
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a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 17.
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4
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1, 65535, 21523360, 2147450880, 38146972656, 1410533397600, 5538821761600, 70367670435840, 308836690967520, 2499961853010960, 4594972986357216, 46220358372556800, 55451384098598320, 362986684146456000
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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^16 such that the quotient group Z^16 / 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_16(n)/J_1(n) = J_16(n)/A000010(n), where J_k is the k-th Jordan totient function.
Multiplicative with a(p^e) = p^(15e-15) * (p^16-1) / (p-1).
For squarefree n, a(n) = A000203(n^15). (End)
Sum_{k=1..n} a(k) ~ c * n^16, where c = (1/16) * Product_{p prime} (1 + (p^15-1)/((p-1)*p^16)) = 0.1214735403... .
Sum_{k>=1} 1/a(k) = zeta(15)*zeta(16) * Product_{p prime} (1 - 2/p^16 + 1/p^31) = 1.00001530597583... . (End)
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MAPLE
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add(numtheory[mobius](n/d)*d^16, d=numtheory[divisors](n)) ;
%/numtheory[phi](n) ;
end proc:
for n from 1 to 5000 do
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MATHEMATICA
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A161167[n_]:=DivisorSum[n, MoebiusMu[n/#]*#^(17-1)/EulerPhi[n]&]; Array[A161167, 20]
f[p_, e_] := p^(15*e - 15) * (p^16-1) / (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^15, d, if(ispower(d, 16), moebius(sqrtnint(d, 16))*sigma(n^15/d), 0))) \\ Altug Alkan, Nov 26 2015
(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i, 1]^16 - 1)*f[i, 1]^(15*f[i, 2] - 15)/(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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