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A161213 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 18.

%I A161213 #43 Apr 03 2025 02:57:08
%S A161213 1,131071,64570081,8589869056,190734863281,8463265086751,
%T A161213 38771752331201,562945658454016,2779530261754401,24999809265103951,
%U A161213 50544702849929377,554648540725313536,720867993281778161,5081852349802846271,12315765571578095761,36893206672442392576
%N A161213 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 18.
%C A161213 a(n) is the number of lattices L in Z^17 such that the quotient group Z^17 / L is C_n. - _Álvar Ibeas_, Nov 26 2015
%H A161213 Enrique Pérez Herrero, <a href="/A161213/b161213.txt">Table of n, a(n) for n = 1..5000</a>
%H A161213 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 A161213 <a href="/index/J#nome">Index to Jordan function ratios J_k/J_1</a>.
%F A161213 a(n) = J_17(n)/A000010(n), where J_17 is the 17th Jordan totient function.
%F A161213 From _Álvar Ibeas_, Nov 26 2015: (Start)
%F A161213 Multiplicative with a(p^e) = p^(16e-16) * (p^17-1) / (p-1).
%F A161213 For squarefree n, a(n) = A000203(n^16). (End)
%F A161213 From _Amiram Eldar_, Nov 08 2022: (Start)
%F A161213 Sum_{k=1..n} a(k) ~ c * n^17, where c = (1/17) * Product_{p prime} (1 + (p^16-1)/((p-1)*p^17)) = 0.1143286202... .
%F A161213 Sum_{k>=1} 1/a(k) = zeta(16)*zeta(17) * Product_{p prime} (1 - 2/p^17 + 1/p^33) = 1.000007645061593... . (End)
%F A161213 a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^17). - _Ridouane Oudra_, Apr 02 2025
%p A161213 A161213 := proc(n)
%p A161213     add(numtheory[mobius](n/d)*d^17,d=numtheory[divisors](n)) ;
%p A161213     %/numtheory[phi](n) ;
%p A161213 end proc:
%p A161213 for n from 1 to 5000 do
%p A161213     printf("%d %d\n",n,A161213(n)) ;
%p A161213 end do: # _R. J. Mathar_, Mar 15 2016
%t A161213 A161213[n_]:=DivisorSum[n,MoebiusMu[n/#]*#^(18-1)/EulerPhi[n]&]; Array[A161213,20]
%t A161213 f[p_, e_] := p^(16*e - 16) * (p^17-1) / (p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 20] (* _Amiram Eldar_, Nov 08 2022 *)
%o A161213 (PARI) A161213(n)=sumdiv(n,d,moebius(n/d)*d^17)/eulerphi(n);
%o A161213 (PARI) vector(100, n, sumdiv(n^16, d, if(ispower(d, 17), moebius(sqrtnint(d, 16))*sigma(n^16/d), 0))) \\ _Altug Alkan_, Nov 26 2015
%o A161213 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^17 - 1)*f[i,1]^(16*f[i,2] - 16)/(f[i,1] - 1));} \\ _Amiram Eldar_, Nov 08 2022
%Y A161213 Column 17 of A263950.
%Y A161213 Cf. A000010, A000203, A008683, A013674, A013675.
%K A161213 nonn,mult
%O A161213 1,2
%A A161213 _N. J. A. Sloane_, Nov 19 2009
%E A161213 Definition corrected by _Enrique Pérez Herrero_, Oct 30 2010