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 A160891 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 5. 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 (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 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. 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 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 *) 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 *) 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 Column 4 of A263950. Cf. A000010, A000203, A059377, A160870. Sequence in context: A321491 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 July 16 08:10 EDT 2024. Contains 374345 sequences. (Running on oeis4.)