This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A160889 #40 Mar 30 2025 14:17:12
%S A160889 1,7,13,28,31,91,57,112,117,217,133,364,183,399,403,448,307,819,381,
%T A160889 868,741,931,553,1456,775,1281,1053,1596,871,2821,993,1792,1729,2149,
%U A160889 1767,3276,1407,2667,2379,3472,1723,5187,1893,3724,3627,3871,2257,5824,2793
%N A160889 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 4.
%C A160889 Dirichlet convolution of A000290 and the series of absolute values of A063441. - _R. J. Mathar_, Jun 20 2011
%C A160889 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 A160889 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 A160889 Enrique Pérez Herrero, <a href="/A160889/b160889.txt">Table of n, a(n) for n = 1..5000</a>
%H A160889 <a href="/index/J#nome">Index to Jordan function ratios J_k/J_1</a>
%F A160889 Moebius transform of A064969. Multiplicative with a(p^e) = (p^2+p+1)*p^(2*e-2). - _Vladeta Jovovic_, Nov 21 2009
%F A160889 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 A160889 Dirichlet g.f.: zeta(s-2)*product_{primes p} (1+p^(1-s)+p^(-s)). - _R. J. Mathar_, Jun 20 2011
%F A160889 From _Álvar Ibeas_, Oct 30 2015: (Start)
%F A160889 a(n) = A254981(n^2). For squarefree n, a(n) = A000203(n^2).
%F A160889 a(n) = Sum_{d|n, n/d squarefree} d^2 * A000203(n/d).
%F A160889 (End)
%F A160889 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 A160889 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
%F A160889 a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^3). - _Ridouane Oudra_, Mar 26 2025
%e A160889 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 A160889 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 A160889[n_]:=DivisorSum[n,MoebiusMu[n/# ]*#^(4-1)/EulerPhi[n]&] (* _Enrique Pérez Herrero_, Aug 22 2010 *)
%o A160889 (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
%Y A160889 Cf. A156304, A000203.
%K A160889 nonn,mult
%O A160889 1,2
%A A160889 _N. J. A. Sloane_, Nov 19 2009
%E A160889 Definition corrected by _Vladeta Jovovic_, Nov 21 2009
%E A160889 Typo in Mathematica program and formula fixed by _Enrique Pérez Herrero_, Oct 19 2010