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%I #14 Oct 25 2022 02:38:01
%S 1,7,3,19,3,21,3,37,6,21,3,57,3,21,9,61,3,42,3,57,9,21,3,111,6,21,10,
%T 57,3,63,3,91,9,21,9,114,3,21,9,111,3,63,3,57,18,21,3,183,6,42,9,57,3,
%U 70,9,111,9,21,3,171,3,21,18,127,9,63,3,57,9,63,3,222,3,21,18,57,9,63,3,183
%N Dirichlet g.f.: (1+4/2^s+1/4^s)*zeta(s)^3.
%H Antti Karttunen, <a href="/A145399/b145399.txt">Table of n, a(n) for n = 1..65537</a>
%H J. S. Rutherford, <a href="https://dx.doi.org/10.1107/S0108767392000898">The enumeration and symmetry-significant properties of derivative lattices</a>, Act. Cryst. A48 (1992), 500-508. Table 1, symmetry Pmmm.
%F From _Amiram Eldar_, Oct 25 2022: (Start):
%F Multiplicative with a(2^e) = 3*e*(e+1)+1 and a(p^e) = (e+1)*(e+2)/2 if p > 2.
%F Sum_{k=1..n} a(k) ~ (13/8)*n*log(n)^2 + c_1*n*log(n) + c_2*n, where c_1 = 39*gamma/4 - 5*log(2)/2 - 13/4 and c_2 = 13/4 + 39*gamma*(gamma-1)/4 - 15*gamma*log(2)/2 - 39*gamma_1/4 + 5*log(2)/2 + 3*log(2)^2/2, where gamma is Euler's constant (A001620) and gamma_1 is the 1st Stieltjes constant (A082633). (End)
%p read("transforms") : nmax := 100 :
%p L := [1,4,0,1,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*(e + 1) + 1;; 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+4*X+X^2,200)
%o t3=dirmul(t1,t2)
%o A145399(n) = t3[n]; \\ This line added by _Antti Karttunen_, Sep 27 2018
%o (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] == 2, 3*f[i,2]*(f[i,2]+1)+1, (f[i,2]+1)*(f[i,2]+2)/2)); } \\ _Amiram Eldar_, Oct 25 2022
%Y Cf. A007425, A145511.
%Y Cf. A001620, A082633.
%K nonn,mult
%O 1,2
%A _N. J. A. Sloane_, Mar 13 2009
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