A161025 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 15.
1, 16383, 2391484, 134209536, 1525878906, 39179682372, 113037178808, 1099444518912, 3812797945332, 24998474116998, 37974983358324, 320959957991424, 328114698808274, 1851888100411464, 3649114989636504
Offset: 1
Links
- Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000
- Jin Ho Kwak and Jaeun Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.
- Index to Jordan function ratios J_k/J_1.
Programs
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Maple
A161025 := proc(n) add(numtheory[mobius](n/d)*d^14,d=numtheory[divisors](n)) ; %/numtheory[phi](n) ; end proc: for n from 1 to 5000 do printf("%d %d\n",n,A161025(n)) ; end do: # R. J. Mathar, Mar 15 2016
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Mathematica
A161025[n_]:=DivisorSum[n,MoebiusMu[n/#]*#^(15-1)/EulerPhi[n]&] f[p_, e_] := p^(13*e - 13) * (p^14-1) / (p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 20] (* Amiram Eldar, Nov 08 2022 *)
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PARI
vector(100, n, sumdiv(n^13, d, if(ispower(d, 14), moebius(sqrtnint(d, 14))*sigma(n^13/d), 0))) \\ Altug Alkan, Nov 26 2015
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PARI
a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^14 - 1)*f[i,1]^(13*f[i,2] - 13)/(f[i,1] - 1));} \\ Amiram Eldar, Nov 08 2022
Formula
a(n) = J_14(n)/J_1(n) where J_14 and J_1(n) = A000010(n) are Jordan functions. - R. J. Mathar, Jul 12 2011
From Álvar Ibeas, Nov 26 2015: (Start)
Multiplicative with a(p^e) = p^(13e-13) * (p^14-1) / (p-1).
For squarefree n, a(n) = A000203(n^13). (End)
From Amiram Eldar, Nov 08 2022: (Start)
Sum_{k=1..n} a(k) ~ c * n^14, where c = (1/14) * Product_{p prime} (1 + (p^13-1)/((p-1)*p^14)) = 0.1388226555... .
Sum_{k>=1} 1/a(k) = zeta(13)*zeta(14) * Product_{p prime} (1 - 2/p^14 + 1/p^27) = 1.00006146517418... . (End)
a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^14). - Ridouane Oudra, Apr 02 2025
Extensions
Definition corrected by Enrique Pérez Herrero, Oct 30 2010
Comments