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%I #39 Nov 08 2022 08:07:13
%S 1,15,40,120,156,600,400,960,1080,2340,1464,4800,2380,6000,6240,7680,
%T 5220,16200,7240,18720,16000,21960,12720,38400,19500,35700,29160,
%U 48000,25260,93600,30784,61440,58560,78300,62400,129600,52060,108600,95200
%N a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 5.
%C 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
%H Enrique Pérez Herrero, <a href="/A160891/b160891.txt">Table of n, a(n) for n = 1..5000</a>
%H Jin Ho Kwak and Jaeun Lee, <a href="https://doi.org/10.1142/9789812799890_0005">Enumeration of graph coverings, surface branched coverings and related group theory</a>, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.
%H <a href="/index/J#nome">Index to Jordan function ratios J_k/J_1</a>.
%F 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
%F Multiplicative with a(p^e) = p^(3e-3)*(1+p+p^2+p^3). - _R. J. Mathar_, Jul 10 2011
%F For squarefree n, a(n) = A000203(n^3). - _Álvar Ibeas_, Oct 30 2015
%F 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
%F 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
%e 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.
%e 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.
%p 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:
%p seq(A160891(n),n=1..20) ; # _R. J. Mathar_, Jul 10 2011
%t A160891[n_]:=DivisorSum[n,MoebiusMu[n/#]*#^(5-1)/EulerPhi[n]&] (* _Enrique Pérez Herrero_, Oct 19 2010 *)
%t 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 *)
%o (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
%o (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
%Y Column 4 of A263950.
%Y Cf. A000010, A000203, A059377, A160870.
%K nonn,mult
%O 1,2
%A _N. J. A. Sloane_, Nov 19 2009
%E Definition corrected by _Enrique Pérez Herrero_, Aug 22 2010
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