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%I #48 Nov 08 2022 08:07:28
%S 1,127,1093,8128,19531,138811,137257,520192,796797,2480437,1948717,
%T 8883904,5229043,17431639,21347383,33292288,25646167,101193219,
%U 49659541,158747968,150021901,247487059,154764793,568569856,305171875,664088461
%N a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 8.
%C 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 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 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 <a href="/index/J#nome">Index to Jordan function ratios J_k/J_1</a>.
%F 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 From _Álvar Ibeas_, Oct 30 2015: (Start)
%F Multiplicative with a(p^e) = p^(6e-6) * (p^7-1) / (p-1).
%F For squarefree n, a(n) = A000203(n^6). (End)
%F From _Amiram Eldar_, Nov 08 2022: (Start)
%F 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 Sum_{k>=1} 1/a(k) = zeta(6)*zeta(7) * Product_{p prime} (1 - 2/p^7 + 1/p^13) = 1.008982290854... . (End)
%p A160897 := proc(n)
%p add(numtheory[mobius](n/d)*d^7,d=numtheory[divisors](n)) ;
%p %/numtheory[phi](n) ;
%p end proc:
%p for n from 1 to 5000 do
%p printf("%d %d\n",n,A160897(n)) ;
%p end do: # _R. J. Mathar_, Mar 14 2016
%t A160897[n_]:=DivisorSum[n, MoebiusMu[n/# ]*#^(8 - 1)/EulerPhi[n] &] (* _Enrique Pérez Herrero_, Oct 27 2010 *)
%t 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 (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 (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 Column 7 of A263950.
%Y Cf. A000010, A000203, A013664, A013665, A069092.
%K nonn,mult
%O 1,2
%A _N. J. A. Sloane_, Nov 19 2009
%E Definition corrected by _Enrique Pérez Herrero_, Oct 27 2010
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