A035191 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 9.

1, 2, 1, 3, 2, 2, 2, 4, 1, 4, 2, 3, 2, 4, 2, 5, 2, 2, 2, 6, 2, 4, 2, 4, 3, 4, 1, 6, 2, 4, 2, 6, 2, 4, 4, 3, 2, 4, 2, 8, 2, 4, 2, 6, 2, 4, 2, 5, 3, 6, 2, 6, 2, 2, 4, 8, 2, 4, 2, 6, 2, 4, 2, 7, 4, 4, 2, 6, 2, 8, 2, 4, 2, 4, 3, 6, 4, 4, 2, 10, 1

Offset

1,2

Comments

Number of divisors of n not congruent to 0 mod 3. - Vladeta Jovovic, Oct 26 2001

a(n) is the number of factors (over Q) of the polynomial x^(2n) + x^n + 1 . a(n) = d(3n) - d(n) where d() is the divisor function. - Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 28 2003

Equals Mobius transform of A011655. - Gary W. Adamson, Apr 24 2009

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Formula

Multiplicative with a(3^e)=1 and a(p^e)=e+1 for p<>3.

G.f.: Sum_{k>0} x^k*(1+x^k)/(1-x^(3*k)). - Vladeta Jovovic, Dec 16 2002

a(n) = A001817(n) + A001822(n). [Reinhard Zumkeller, Nov 26 2011]

a(n) = tau(3*n) - tau(n). - Ridouane Oudra, Sep 05 2020

From Amiram Eldar, Nov 27 2022: (Start)

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

Sum_{k=1..n} a(k) ~ (2*n*log(n) + (4*gamma + log(3) - 2)*n)/3, where gamma is Euler's constant (A001620). (End)

a(n) = Sum_{d|n} Kronecker(9, d). - Amiram Eldar, Nov 20 2023

Maple

for n from 1 to 500 do a := ifactors(n):s := 1:for k from 1 to nops(a[2]) do p := a[2][k][1]:e := a[2][k][2]: if p=3 then b := 1:else b := e+1:fi:s := s*b:od:printf(`%d, `, s); od:
# alternative
A035191 := proc(n)
    A001817(n)+A001822(n) ;
end proc:
[seq(A035191(n), n=1..100)] ; # R. J. Mathar, Sep 25 2017

Mathematica

a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[9, #] &]]; Table[ a[n], {n, 1, 100}] (* G. C. Greubel, Apr 27 2018 *)
f[3, e_] := 1; f[p_, e_] := e+1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 26 2020 *)

Prog

(PARI) my(m=9); direuler(p=2, 101, 1/(1-(kronecker(m, p)*(X-X^2))-X))
(PARI) a(n) = sumdiv(n, d, kronecker(9, d)); \\ Amiram Eldar, Nov 20 2023
(Haskell)
a035191 n = a001817 n + a001822 n  -- Reinhard Zumkeller, Nov 26 2011
(Magma) [NumberOfDivisors(n)/Valuation(3*n, 3): n in [1..100]]; // Vincenzo Librandi, Jun 03 2019

Crossrefs

Cf. A011655, A035207, A046913, A054584.

Cf. A000005, A001227, A001620, A001817, A001817.

Sequence in context: A279027 A103151 A035221 * A297167 A325224 A303389

Adjacent sequences: A035188 A035189 A035190 * A035192 A035193 A035194

Keyword

nonn,mult,easy

Author

N. J. A. Sloane

Status

approved