%I #16 Oct 25 2022 02:37:30
%S 1,3,3,7,3,9,3,13,6,9,3,21,3,9,9,25,3,18,3,21,9,9,3,39,6,9,10,21,3,27,
%T 3,43,9,9,9,42,3,9,9,39,3,27,3,21,18,9,3,75,6,18,9,21,3,30,9,39,9,9,3,
%U 63,3,9,18,67,9,27,3,21,9,27,3,78,3,9,18,21,9,27,3,75,15,9,3,63,9,9,9,39,3
%N Dirichlet g.f.: (1+1/4^s+4/16^s)*zeta(s)^3.
%C Dirichlet convolution of [1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,0,0,...] with A007425.
%H Antti Karttunen, <a href="/A145501/b145501.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, Bravais lattice / symmetry Immm.
%F From _Amiram Eldar_, Oct 25 2022: (Start):
%F Multiplicative with a(2) = 3, a(2^e) = 3*(e-2)*(e-1)+7 for e > 1, and a(p^e) = (e+1)*(e+2)/2 if p > 2.
%F Sum_{k=1..n} a(k) ~ (3/4)*n*log(n)^2 + c_1*n*log(n) + c_2*n, where c_1 = 9*gamma/2 - 3*log(2)/2 - 3/2 and c_2 = 3/2 + 9*gamma*(gamma-1)/2 - 9*gamma*log(2)/2 - 9*gamma_1/2 + 3*log(2)/2 + 5*log(2)^2/2, where gamma is Euler's constant (A001620) and gamma_1 is the 1st Stieltjes constant (A082633). (End)
%p nmax := 100 :
%p L := [1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,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, 1] = 3; f[2, e_] := 3*(e - 2)*(e - 1) + 7; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Oct 25 2022 *)
%o (PARI)
%o up_to = 10000
%o t1=direuler(p=2, up_to, 1/(1-X)^3);
%o t2=direuler(p=2, 2, 1+1*X^2+4*X^4, up_to);
%o t3=dirmul(t1, t2);
%o \\ _Antti Karttunen_, Sep 24 2017, after PARI-code in A145444.
%o (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] == 2, if(f[i,2] == 1, 3, 3*(f[i,2]-2)*(f[i,2]-1)+7), (f[i,2]+1)*(f[i,2]+2)/2)); } \\ _Amiram Eldar_, Oct 25 2022
%Y Cf. A007425, A145444.
%Y Cf. A001620, A082633.
%K nonn,mult
%O 1,2
%A _N. J. A. Sloane_, Mar 14 2009