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A160953 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 10.

%I A160953 #36 Apr 01 2025 19:49:38
%S A160953 1,511,9841,130816,488281,5028751,6725601,33488896,64566801,249511591,
%T A160953 235794769,1287360256,883708281,3436782111,4805173321,8573157376,
%U A160953 7411742281,32993635311,17927094321,63874967296,66186639441,120491126959,81870575521,329564225536
%N A160953 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 10.
%C A160953 a(n) is the number of lattices L in Z^9 such that the quotient group Z^9 / L is C_n. - _Álvar Ibeas_, Nov 03 2015
%H A160953 Enrique Pérez Herrero, <a href="/A160953/b160953.txt">Table of n, a(n) for n = 1..5000</a>
%H A160953 Enrique Pérez Herrero, <a href="https://sites.google.com/site/psychgeom/psychgeom/JordanTotientFunction.m?attredirects=0&amp;d=1">Mathematica Package: Jordan Totient Function</a>.
%H A160953 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 A160953 <a href="/index/J#nome">Index to Jordan function ratios J_k/J_1</a>.
%F A160953 a(n) = J_9(n)/phi(n) = A069094(n)/A000010(n).
%F A160953 From _Álvar Ibeas_, Nov 03 2015: (Start)
%F A160953 Multiplicative with a(p^e) = p^(8e-8) * (p^9-1) / (p-1).
%F A160953 For squarefree n, a(n) = A000203(n^8). (End)
%F A160953 From _Amiram Eldar_, Nov 08 2022: (Start)
%F A160953 Sum_{k=1..n} a(k) ~ c * n^9, where c = (1/9) * Product_{p prime} (1 + (p^8-1)/((p-1)*p^9)) = 0.2156692448... .
%F A160953 Sum_{k>=1} 1/a(k) = zeta(8)*zeta(9) * Product_{p prime} (1 - 2/p^9 + 1/p^17) = 1.002068659133... . (End)
%F A160953 a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^9). - _Ridouane Oudra_, Apr 01 2025
%p A160953 A160953 := proc(n)
%p A160953     add(numtheory[mobius](n/d)*d^9,d=numtheory[divisors](n)) ;
%p A160953     %/numtheory[phi](n) ;
%p A160953 end proc:
%p A160953 for n from 1 to 5000 do
%p A160953     printf("%d %d\n",n,A160953(n)) ;
%p A160953 end do: # _R. J. Mathar_, Mar 14 2016
%t A160953 JordanTotient[n_,k_:1]:=DivisorSum[n,#^k*MoebiusMu[n/#]&]/;(n>0)&&IntegerQ[n];
%t A160953 A160953[n_]:=JordanTotient[n,9]/JordanTotient[n];
%t A160953 f[p_, e_] := p^(8*e - 8) * (p^9-1) / (p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 25] (* _Amiram Eldar_, Nov 08 2022 *)
%o A160953 (PARI) vector(100, n, sumdiv(n^8, d, if(ispower(d, 9), moebius(sqrtnint(d, 9))*sigma(n^8/d), 0))) \\ _Altug Alkan_, Nov 05 2015
%o A160953 (PARI) a(n) = {f = factor(n); for (i=1, #f~, p = f[i,1]; f[i,1] = p^(8*f[i,2]-8)*(p^9-1)/(p-1); f[i,2] = 1;); factorback(f);} \\ _Michel Marcus_, Nov 12 2015
%Y A160953 Column 9 of A263950.
%Y A160953 Cf. A000010, A000203, A008683, A013666, A013667, A069094.
%K A160953 nonn,mult
%O A160953 1,2
%A A160953 _N. J. A. Sloane_, Nov 19 2009
%E A160953 Definition corrected by _Enrique Pérez Herrero_, Oct 30 2010