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A160891 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 5. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

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

Index to Jordan function ratios J_k/J_1

FORMULA

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

EXAMPLE

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.

MAPLE

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

MATHEMATICA

A160891[n_]:=DivisorSum[n, MoebiusMu[n/#]*#^(5-1)/EulerPhi[n]&] (* Enrique Pérez Herrero, Oct 19 2010 *)

PROG

(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

CROSSREFS

Sequence in context: A044473 A067724 A005337 * A223425 A175926 A038991

Adjacent sequences:  A160888 A160889 A160890 * A160892 A160893 A160894

KEYWORD

nonn,mult

AUTHOR

N. J. A. Sloane, Nov 19 2009

EXTENSIONS

Definition corrected by Enrique Pérez Herrero, Aug 22 2010

STATUS

approved

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Last modified June 28 14:25 EDT 2016. Contains 274266 sequences.