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%I #17 Oct 25 2022 02:36:47
%S 1,3,3,9,3,9,3,21,6,9,3,27,3,9,9,39,3,18,3,27,9,9,3,63,6,9,10,27,3,27,
%T 3,63,9,9,9,54,3,9,9,63,3,27,3,27,18,9,3,117,6,18,9,27,3,30,9,63,9,9,
%U 3,81,3,9,18,93,9,27,3,27,9,27,3,126,3,9,18,27,9,27,3,117,15,9,3,81,9,9,9,63
%N Dirichlet g.f.: (1+3/4^s+2/8^s)*zeta(s)^3.
%C Dirichlet convolution of [1,0,0,3,0,0,0,2,0,0,...] with A007425. - _R. J. Mathar_, Sep 25 2017
%H Antti Karttunen, <a href="/A145444/b145444.txt">Table of n, a(n) for n = 1..10000</a>
%H J. S. Rutherford, <a href="https://doi.org/10.1107/S0108767392000898">The enumeration and symmetry-significant properties of derivative lattices</a>, Acta Cryst. A48 (1992), 500-508. See Table 1, symmetry Cmmm.
%F From _Amiram Eldar_, Oct 25 2022: (Start):
%F Multiplicative with a(2^e) = 3*(e-1)*e+3 for e > 0, and a(p^e) = (e+1)*(e+2)/2 if p > 2.
%F Sum_{k=1..n} a(k) ~ n*log(n)^2 + c_1*n*log(n) + c_2*n, where c_1 = 6*gamma - 9*log(2)/4 - 2 and c_2 = 2 + 6*gamma*(gamma-1) - 27*gamma*log(2)/4 - 6*gamma_1 + 9*log(2)/4 + 21*log(2)^2/8, where gamma is Euler's constant (A001620) and gamma_1 is the 1st Stieltjes constant (A082633). (End)
%p nmax := 10000 :
%p L := [1,0,0,3,0,0,0,2,seq(0,i=1..nmax)] :
%p MOBIUSi(%) :
%p MOBIUSi(%) :
%p MOBIUSi(%) ; # _R. J. Mathar_, Sep 25 2017
%t f[p_, e_] := (e + 1)*(e + 2)/2; f[2, e_] := 3*(e - 1)*e + 3; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Oct 25 2022 *)
%o (PARI) t1=direuler(p=2,200,1/(1-X)^3)
%o t2=direuler(p=2,2,1+3*X^2+2*X^3,200)
%o t3=dirmul(t1,t2)
%o (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] == 2, 3*(f[i,2]-1)*f[i,2]+3, (f[i,2]+1)*(f[i,2]+2)/2)); } \\ _Amiram Eldar_, Oct 25 2022
%Y Cf. A007425, A145501.
%Y Cf. A001620, A082633.
%K nonn,mult
%O 1,2
%A _N. J. A. Sloane_, Mar 14 2009
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