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A160889 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 4. 7
1, 7, 13, 28, 31, 91, 57, 112, 117, 217, 133, 364, 183, 399, 403, 448, 307, 819, 381, 868, 741, 931, 553, 1456, 775, 1281, 1053, 1596, 871, 2821, 993, 1792, 1729, 2149, 1767, 3276, 1407, 2667, 2379, 3472, 1723, 5187, 1893, 3724, 3627, 3871, 2257, 5824, 2793 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Dirichlet convolution of A000290 and the series of absolute values of A063441. - R. J. Mathar, Jun 20 2011
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
REFERENCES
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.
LINKS
Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000
FORMULA
Moebius transform of A064969. Multiplicative with a(p^e) = (p^2+p+1)*p^(2*e-2). - Vladeta Jovovic, Nov 21 2009
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
Dirichlet g.f.: zeta(s-2)*product_{primes p} (1+p^(1-s)+p^(-s)). - R. J. Mathar, Jun 20 2011
From Álvar Ibeas, Oct 30 2015: (Start)
a(n) = A254981(n^2). For squarefree n, a(n) = A000203(n^2).
a(n) = Sum_{d|n, n/d squarefree} d^2 * A000203(n/d).
(End)
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
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
EXAMPLE
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.
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).
MATHEMATICA
A160889[n_]:=DivisorSum[n, MoebiusMu[n/# ]*#^(4-1)/EulerPhi[n]&] (* Enrique Pérez Herrero, Aug 22 2010 *)
PROG
(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
CROSSREFS
Sequence in context: A146718 A146646 A096194 * A360357 A283650 A045463
KEYWORD
nonn,mult
AUTHOR
N. J. A. Sloane, Nov 19 2009
EXTENSIONS
Definition corrected by Vladeta Jovovic, Nov 21 2009
Typo in Mathematica program and formula fixed by Enrique Pérez Herrero, Oct 19 2010
STATUS
approved

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Last modified March 29 00:26 EDT 2024. Contains 371264 sequences. (Running on oeis4.)