A145501 Dirichlet g.f.: (1+1/4^s+4/16^s)*zeta(s)^3.

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, 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, 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

Offset

1,2

Comments

Dirichlet convolution of [1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,0,0,...] with A007425.

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, Bravais lattice / symmetry Immm.

Formula

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

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.

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)

Maple

nmax := 100 :
L := [1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 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, 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 *)

Prog

(PARI)
up_to = 10000
t1=direuler(p=2, up_to, 1/(1-X)^3);
t2=direuler(p=2, 2, 1+1*X^2+4*X^4, up_to);
t3=dirmul(t1, t2);
\\ Antti Karttunen, Sep 24 2017, after PARI-code in A145444.
(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

Crossrefs

Cf. A007425, A145444.

Cf. A001620, A082633.

Sequence in context: A030316 A208884 A034257 * A370296 A182139 A092693

Adjacent sequences: A145498 A145499 A145500 * A145502 A145503 A145504

Keyword

nonn,mult

Author

N. J. A. Sloane, Mar 14 2009

Status

approved