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A160891
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a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 5.
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4
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1, 15, 40, 120, 156, 600, 400, 960, 1080, 2340, 1464, 4800, 2380, 6000, 6240, 7680, 5220, 16200, 7240, 18720, 16000, 21960, 12720, 38400, 19500, 35700, 29160, 48000, 25260, 93600, 30784, 61440, 58560, 78300, 62400, 129600, 52060, 108600, 95200
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OFFSET
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1,2
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COMMENTS
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a(n) is the number of lattices L in Z^4 such that the quotient group Z^4 / L is C_nm x (C_m)^3 (and also (C_nm)^3 x C_m), for every m>=1. - Álvar Ibeas, Oct 30 2015
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REFERENCES
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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.
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LINKS
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Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000
Index to Jordan function ratios J_k/J_1
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FORMULA
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a(n) = J_4(n)/J_1(n) = J_4(n)/phi(n) = A059377(n)/A000010(n), where J_k is the k-th Jordan Totient Function. - Enrique Pérez Herrero, Oct 19 2010
Multiplicative with a(p^e) = p^(3e-3)*(1+p+p^2+p^3). - R. J. Mathar, Jul 10 2011
For squarefree n, a(n) = A000203(n^3). - Álvar Ibeas, Oct 30 2015
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^3/((p^3-1)*(p^3+p^2+p+1))) = 1.115923965261131974852254388404911045036763705978837384729819264463715993... - Vaclav Kotesovec, Sep 20 2020
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EXAMPLE
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There are 1395 = A160870(8,4) lattices of volume 8 in Z^4. Among them, a(8) = 960 give the quotient group C_8 and a(2) = 15 give C_2 x C_2 x C_2.
Among the lattices of volume 64 in Z^4, there are a(4) = 120 such that the quotient group is C_4 x C_4 x C_4 and other 120 with quotient group C_8 x (C_2)^3.
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MAPLE
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A160891 := proc(n) a := 1 ; for f in ifactors(n)[2] do p := op(1, f) ; e := op(2, f) ; a := a*p^(3*e-3)*(1+p+p^2+p^3) ; end do; a; end proc:
seq(A160891(n), n=1..20) ; # R. J. Mathar, Jul 10 2011
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MATHEMATICA
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A160891[n_]:=DivisorSum[n, MoebiusMu[n/#]*#^(5-1)/EulerPhi[n]&] (* Enrique Pérez Herrero, Oct 19 2010 *)
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PROG
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(PARI) vector(50, n, sumdiv(n^3, d, if(ispower(d, 4), moebius(sqrtnint(d, 4))*sigma(n^3/d), 0))) \\ Altug Alkan, Oct 30 2014
(PARI) a(n) = {f = factor(n); for (i=1, #f~, p = f[i, 1]; f[i, 1] = p^(3*f[i, 2]-3)*(1+p+p^2+p^3); f[i, 2] = 1; ); factorback(f); } \\ Michel Marcus, Nov 12 2015
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CROSSREFS
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Sequence in context: A321491 A067724 A005337 * A223425 A175926 A038991
Adjacent sequences: A160888 A160889 A160890 * A160892 A160893 A160894
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KEYWORD
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nonn,mult
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AUTHOR
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N. J. A. Sloane, Nov 19 2009
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EXTENSIONS
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Definition corrected by Enrique Pérez Herrero, Aug 22 2010
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STATUS
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approved
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