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A160897 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 8.

%I A160897 #53 Apr 01 2025 19:48:19
%S A160897 1,127,1093,8128,19531,138811,137257,520192,796797,2480437,1948717,
%T A160897 8883904,5229043,17431639,21347383,33292288,25646167,101193219,
%U A160897 49659541,158747968,150021901,247487059,154764793,568569856,305171875,664088461,580865013,1115624896
%N A160897 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 8.
%C A160897 a(n) is the number of lattices L in Z^7 such that the quotient group Z^7 / L is C_n. - _Álvar Ibeas_, Oct 30 2015
%H A160897 Enrique Pérez Herrero and Robert Israel, <a href="/A160897/b160897.txt">Table of n, a(n) for n = 1..10000</a>  (n = 1..5000 from Enrique Pérez Herrero)
%H A160897 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 A160897 <a href="/index/J#nome">Index to Jordan function ratios J_k/J_1</a>.
%F A160897 a(n) = J_7(n)/J_1(n) = J_7(n)/phi(n) = A069092(n)/A000010(n), where J_k is the k-th Jordan totient function. - _Enrique Pérez Herrero_, Oct 27 2010
%F A160897 From _Álvar Ibeas_, Oct 30 2015: (Start)
%F A160897 Multiplicative with a(p^e) = p^(6e-6) * (p^7-1) / (p-1).
%F A160897 For squarefree n, a(n) = A000203(n^6). (End)
%F A160897 From _Amiram Eldar_, Nov 08 2022: (Start)
%F A160897 Sum_{k=1..n} a(k) ~ c * n^7, where c = (1/7) * Product_{p prime} (1 + (p^6-1)/((p-1)*p^7)) = 0.2761554804... .
%F A160897 Sum_{k>=1} 1/a(k) = zeta(6)*zeta(7) * Product_{p prime} (1 - 2/p^7 + 1/p^13) = 1.008982290854... . (End)
%F A160897 a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^7). - _Ridouane Oudra_, Apr 01 2025
%p A160897 A160897 := proc(n)
%p A160897     add(numtheory[mobius](n/d)*d^7,d=numtheory[divisors](n)) ;
%p A160897     %/numtheory[phi](n) ;
%p A160897 end proc:
%p A160897 for n from 1 to 5000 do
%p A160897     printf("%d %d\n",n,A160897(n)) ;
%p A160897 end do: # _R. J. Mathar_, Mar 14 2016
%t A160897 A160897[n_]:=DivisorSum[n, MoebiusMu[n/# ]*#^(8 - 1)/EulerPhi[n] &] (* _Enrique Pérez Herrero_, Oct 27 2010 *)
%t A160897 f[p_, e_] := p^(6*e - 6) * (p^7-1) / (p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 25] (* _Amiram Eldar_, Nov 08 2022 *)
%o A160897 (PARI) vector(30, n, sumdiv(n^6, d, if(ispower(d, 7), moebius(sqrtnint(d, 7))*sigma(n^6/d), 0))) \\ _Altug Alkan_, Oct 30 2015
%o A160897 (PARI) a(n) = {f = factor(n); for (i=1, #f~, p = f[i, 1]; f[i,1] = p^(6*f[i,2]-6)*(1+p+p^2+p^3+p^4+p^5+p^6); f[i,2] = 1;); factorback(f);} \\ _Michel Marcus_, Nov 12 2015
%Y A160897 Column 7 of A263950.
%Y A160897 Cf. A000010, A000203, A008683, A013664, A013665, A069092.
%K A160897 nonn,mult
%O A160897 1,2
%A A160897 _N. J. A. Sloane_, Nov 19 2009
%E A160897 Definition corrected by _Enrique Pérez Herrero_, Oct 27 2010