A145444 Dirichlet g.f.: (1+3/4^s+2/8^s)*zeta(s)^3.

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, 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, 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

Offset: 1

Author

N. J. A. Sloane, Mar 14 2009

Keywords

• nonn
• mult

Comments

Dirichlet convolution of [1,0,0,3,0,0,0,2,0,0,...] with A007425. - R. J. Mathar, Sep 25 2017

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• J. S. Rutherford, The enumeration and symmetry-significant properties of derivative lattices, Acta Cryst. A48 (1992), 500-508. See Table 1, symmetry Cmmm.

Crossrefs

Cf. A007425, A145501.

Cf. A001620, A082633.

Programs

Maple

nmax := 10000 :
L := [1,0,0,3,0,0,0,2,seq(0,i=1..nmax)] :
MOBIUSi(%) :
MOBIUSi(%) :
MOBIUSi(%) ; # R. J. Mathar, Sep 25 2017

Mathematica

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 *)

PARI

t1=direuler(p=2,200,1/(1-X)^3)
t2=direuler(p=2,2,1+3*X^2+2*X^3,200)
t3=dirmul(t1,t2)

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

Formula

From Amiram Eldar, Oct 25 2022: (Start):

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.

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)