A160891 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 5.

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

Offset: 1

Author

N. J. A. Sloane, Nov 19 2009

Keywords

• nonn
• mult

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

Examples

			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.
		

Links

• Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000
• Jin Ho Kwak and Jaeun 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.
• Index to Jordan function ratios J_k/J_1.

Crossrefs

Column 4 of A263950.

Cf. A000010, A000203, A059377, A160870.

Programs

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 *)
f[p_, e_] := p^(3 e - 3)*(1 + p + p^2 + p^3); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 40] (* Amiram Eldar, Nov 08 2022 *)

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

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

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

Sum_{k=1..n} a(k) ~ c * n^4, where c = (1/4) * Product_{p prime} (1 + 1/p^2 + 1/p^3 + 1/p^4) = 0.4629765396... . - Amiram Eldar, Nov 08 2022

a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^4). - Ridouane Oudra, Apr 01 2025

Extensions

Definition corrected by Enrique Pérez Herrero, Aug 22 2010