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A161139 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 16.

%I A161139 #39 Apr 02 2025 17:14:01
%S A161139 1,32767,7174453,536854528,7629394531,235085301451,791260251657,
%T A161139 8795824586752,34315186290957,249992370597277,417724816941565,
%U A161139 3851637578973184,4265491084507563,25927224666044919,54736732481116543
%N A161139 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 16.
%C A161139 a(n) is the number of lattices L in Z^15 such that the quotient group Z^15 / L is C_n. - _Álvar Ibeas_, Nov 26 2015
%H A161139 Enrique Pérez Herrero, <a href="/A161139/b161139.txt">Table of n, a(n) for n = 1..5000</a>
%H A161139 Jin Ho Kwak and Jaeun Lee, <a href="https://doi.org/10.1142/9789812799890_0005">Enumeration of graph coverings, surface branched coverings and related group theory</a>, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.
%H A161139 <a href="/index/J#nome">Index to Jordan function ratios J_k/J_1</a>.
%F A161139 a(n) = J_15(n)/J_1(n), where J_15 and J_1(n) = A000010(n) are Jordan functions. - _R. J. Mathar_, Jul 12 2011
%F A161139 From _Álvar Ibeas_, Nov 26 2015: (Start)
%F A161139 Multiplicative with a(p^e) = p^(14e-14) * (p^15-1) / (p-1).
%F A161139 For squarefree n, a(n) = A000203(n^14). (End)
%F A161139 From _Amiram Eldar_, Nov 08 2022: (Start)
%F A161139 Sum_{k=1..n} a(k) ~ c * n^15, where c = (1/15) * Product_{p prime} (1 + (p^14-1)/((p-1)*p^15)) = 0.1295704557... .
%F A161139 Sum_{k>=1} 1/a(k) = zeta(14)*zeta(15) * Product_{p prime} (1 - 2/p^15 + 1/p^29) = 1.00003065989236... . (End)
%F A161139 a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^15). - _Ridouane Oudra_, Apr 02 2025
%p A161139 A161139 := proc(n)
%p A161139     add(numtheory[mobius](n/d)*d^15,d=numtheory[divisors](n)) ;
%p A161139     %/numtheory[phi](n) ;
%p A161139 end proc:
%p A161139 for n from 1 to 5000 do
%p A161139     printf("%d %d\n",n,A161139(n)) ;
%p A161139 end do: # _R. J. Mathar_, Mar 15 2016
%t A161139 A161139[n_] := DivisorSum[n, MoebiusMu[n/#]*#^(16 - 1)/EulerPhi[n] &]; Array[A161139,20] (* _Enrique Pérez Herrero_, Mar 02 2011 *)
%t A161139 f[p_, e_] := p^(14*e - 14) * (p^15-1) / (p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 20] (* _Amiram Eldar_, Nov 08 2022 *)
%o A161139 (PARI) vector(100, n, sumdiv(n^14, d, if(ispower(d, 15), moebius(sqrtnint(d, 15))*sigma(n^14/d), 0))) \\ _Altug Alkan_, Nov 26 2015
%o A161139 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^15 - 1)*f[i,1]^(14*f[i,2] - 14)/(f[i,1] - 1));} \\ _Amiram Eldar_, Nov 08 2022
%Y A161139 Column 15 of A263950.
%Y A161139 Cf. A000010, A000203. A013672, A013673.
%K A161139 nonn,mult
%O A161139 1,2
%A A161139 _N. J. A. Sloane_, Nov 19 2009
%E A161139 Definition corrected by _Enrique Pérez Herrero_, Oct 30 2010