A145511 Dirichlet g.f.: (1-2/2^s+7/4^s)*zeta(s)^3.

1, 1, 3, 7, 3, 3, 3, 19, 6, 3, 3, 21, 3, 3, 9, 37, 3, 6, 3, 21, 9, 3, 3, 57, 6, 3, 10, 21, 3, 9, 3, 61, 9, 3, 9, 42, 3, 3, 9, 57, 3, 9, 3, 21, 18, 3, 3, 111, 6, 6, 9, 21, 3, 10, 9, 57, 9, 3, 3, 63, 3, 3, 18, 91, 9, 9, 3, 21, 9, 9, 3, 114, 3, 3, 18, 21, 9, 9, 3, 111, 15, 3, 3, 63, 9, 3, 9, 57, 3, 18, 9

Offset

1,3

Comments

Dirichlet convolution of [1,-2,0,7,0,0,0,0,...] and A007425. - R. J. Mathar, Feb 07 2011

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

J. S. Rutherford, The enumeration and symmetry-significant properties of derivative lattices, Acta Cryst. A48 (1992), 500-508. See Table 1, Symmetry Fmmm.

Formula

From Amiram Eldar, Oct 25 2022: (Start)

Multiplicative with a(2^e) = 3*(e-1)*e+1 and a(p^e) = (e+1)*(e+2)/2 if p > 2.

Sum_{k=1..n} a(k) ~ (7/8)*n*log(n)^2 + c_1*n*log(n) + c_2*n, where c_1 = 21*gamma/4 - 5*log(2)/2 - 7/4 and c_2 = 7/4 + 21*gamma*(gamma-1)/4 - 15*gamma*log(2)/2 - 21*gamma_1/4 + 5*log(2)/2 + 3*log(2)^2, where gamma is Euler's constant (A001620) and gamma_1 is the 1st Stieltjes constant (A082633). (End)

Maple

read("transforms") :
nmax := 100 :
L := [1, -2, 0, 7, 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 + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 25 2022 *)

Prog

(PARI)
up_to = 65537;
t1 = direuler(p=2, up_to, 1/(1-X)^3);
t3 = direuler(p=2, 2, 1-2*X^1+7*X^2, up_to);
v145511 = dirmul(t1, t3);
A145511(n) = v145511[n]; \\ Antti Karttunen, Sep 27 2018, after R. J. Mathar's PARI-code for A158327
(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 1] == 2, 3*(f[i, 2]-1)*f[i, 2]+1, (f[i, 2]+1)*(f[i, 2]+2)/2)); } \\ Amiram Eldar, Oct 25 2022

Crossrefs

Cf. A007425, A145399, A145444, A145501, A158327, A158360.

Cf. A001620, A082633.

Sequence in context: A010623 A283245 A066065 * A242461 A065443 A198350

Adjacent sequences: A145508 A145509 A145510 * A145512 A145513 A145514

Keyword

nonn,mult

Author

N. J. A. Sloane, Mar 14 2009

Status

approved