A145399 Dirichlet g.f.: (1+4/2^s+1/4^s)*zeta(s)^3.

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, 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, 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

Offset

1,2

Links

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

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

Formula

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

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

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)

Maple

read("transforms") : nmax := 100 :
L := [1, 4, 0, 1, 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*(e + 1) + 1;; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 25 2022 *)

Prog

(PARI) t1=direuler(p=2, 200, 1/(1-X)^3)
t2=direuler(p=2, 2, 1+4*X+X^2, 200)
t3=dirmul(t1, t2)
A145399(n) = t3[n]; \\ This line added by Antti Karttunen, Sep 27 2018
(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

Crossrefs

Cf. A007425, A145511.

Cf. A001620, A082633.

Sequence in context: A283289 A357200 A284477 * A356890 A160797 A050013

Adjacent sequences: A145396 A145397 A145398 * A145400 A145401 A145402

Keyword

nonn,mult

Author

N. J. A. Sloane, Mar 13 2009

Status

approved