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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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5
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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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LINKS
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FORMULA
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Multiplicative with a(p^e) = p^(3e-3)*(1+p+p^2+p^3). - R. J. Mathar, Jul 10 2011
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
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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:
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MATHEMATICA
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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 *)
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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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KEYWORD
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nonn,mult
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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