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a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 4.
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%I #34 Sep 19 2020 12:43:24

%S 1,7,13,28,31,91,57,112,117,217,133,364,183,399,403,448,307,819,381,

%T 868,741,931,553,1456,775,1281,1053,1596,871,2821,993,1792,1729,2149,

%U 1767,3276,1407,2667,2379,3472,1723,5187,1893,3724,3627,3871,2257,5824,2793

%N a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 4.

%C Dirichlet convolution of A000290 and the series of absolute values of A063441. - _R. J. Mathar_, Jun 20 2011

%C a(n) is the number of lattices L in Z^3 such that the quotient group Z^3 / L is C_nm x C_m x C_m (and also C_nm x C_nm x C_m), for every m>=1. - _Álvar Ibeas_, Oct 30 2015

%D J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.

%H Enrique Pérez Herrero, <a href="/A160889/b160889.txt">Table of n, a(n) for n = 1..5000</a>

%H <a href="/index/J#nome">Index to Jordan function ratios J_k/J_1</a>

%F Moebius transform of A064969. Multiplicative with a(p^e) = (p^2+p+1)*p^(2*e-2). - _Vladeta Jovovic_, Nov 21 2009

%F a(n) = J_3(n)/J_1(n)=J_3(n)/phi(n)=A059376(n)/A000010(n), where J_k is the k-th Jordan Totient Function. - _Enrique Pérez Herrero_, Aug 22 2010

%F Dirichlet g.f.: zeta(s-2)*product_{primes p} (1+p^(1-s)+p^(-s)). - _R. J. Mathar_, Jun 20 2011

%F From _Álvar Ibeas_, Oct 30 2015: (Start)

%F a(n) = A254981(n^2). For squarefree n, a(n) = A000203(n^2).

%F a(n) = Sum_{d|n, n/d squarefree} d^2 * A000203(n/d).

%F (End)

%F Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = A330595 = Product_{primes p} (1 + 1/p^2 + 1/p^3) = 1.748932997843245303033906997685114802259883493595480897273662144... - _Vaclav Kotesovec_, Dec 18 2019

%F Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^2/((p^2-1) * (p^2 + p + 1))) = 1.400940662893945919882073637564538872630336562726971915578687405304250550... - _Vaclav Kotesovec_, Sep 19 2020

%e There are 35 = A160870(4,3) lattices of volume 4 in Z^3. Among them, 28 give the quotient group C_4 and 7 give the quotient group C_2 x C_2. Hence, a(4) = 28 and a(2) = 7.

%e There are 2667 = A160870(32,3) lattices of volume 32 in Z^3. Among them, a(32) = 1792 give the quotient group C_32 (m=1); a(4) = 28 give C_8 x C_2 x C_2 (m=2); a(2) = 7 give C_4 x C_4 x C_2 (m=2).

%t A160889[n_]:=DivisorSum[n,MoebiusMu[n/# ]*#^(4-1)/EulerPhi[n]&] (* _Enrique Pérez Herrero_, Aug 22 2010 *)

%o (PARI) vector(100, n, sumdiv(n^2, d, if (ispower(d, 3), moebius(sqrtnint(d, 3))*sigma(n^2/d), 0))) \\ _Altug Alkan_, Oct 30 2015

%K nonn,mult

%O 1,2

%A _N. J. A. Sloane_, Nov 19 2009

%E Definition corrected by _Vladeta Jovovic_, Nov 21 2009

%E Typo in Mathematica program and formula fixed by _Enrique Pérez Herrero_, Oct 19 2010