A327625 Expansion of Sum_{k>=0} x^(3^k) / (1 - x^(3^k))^2.

1, 2, 4, 4, 5, 8, 7, 8, 13, 10, 11, 16, 13, 14, 20, 16, 17, 26, 19, 20, 28, 22, 23, 32, 25, 26, 40, 28, 29, 40, 31, 32, 44, 34, 35, 52, 37, 38, 52, 40, 41, 56, 43, 44, 65, 46, 47, 64, 49, 50, 68, 52, 53, 80, 55, 56, 76, 58, 59, 80, 61, 62, 91, 64, 65, 88, 67, 68, 92, 70

Offset: 1

Author

Ilya Gutkovskiy, Sep 19 2019

Keywords

• nonn
• mult

Comments

Sum of divisors d of n such that n/d is power of 3.

Inverse Moebius transform of A195459.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A000010, A000244, A001651 (fixed points), A051064, A129527, A195459.

Programs

Magma

[(1/2)*&+[EulerPhi(3*d) :d in Divisors(n)]:n in [1..70]]; // Marius A. Burtea, Sep 19 2019

Mathematica

nmax = 70; CoefficientList[Series[Sum[x^(3^k)/(1 - x^(3^k))^2, {k, 0, Floor[Log[3, nmax]] + 1}], {x, 0, nmax}], x] // Rest
a[n_] := DivisorSum[n, # &, IntegerQ[Log[3, n/#]] &]; Table[a[n], {n, 1, 70}]
a[n_] := 1/2 Sum[EulerPhi[3 d], {d, Divisors[n]}]; Table[a[n], {n, 1, 70}]

PARI

A327625(n) = (n+sumdiv(n,d,my(b=0); if(isprimepower(n/d,&b)&&(3==b),d,0))); \\ Antti Karttunen, Sep 19 2019

Formula

G.f. A(x) satisfies: A(x) = A(x^3) + x/(1 - x)^2.

G.f.: Sum_{k>=1} phi(3*k) * x^k / (2 * (1 - x^k)), where phi = A000010.

a(n) = (1/2) * Sum_{d|n} phi(3*d).

From Amiram Eldar, Nov 17 2022: (Start)

Multiplicative with a(3^e) = (3^(e+1)-1)/2, and a(p^e) = p^e for p != 3.

Sum_{k=1..n} a(k) ~ (9/16) * n^2. (End)

Dirichlet g.f.: zeta(s-1)*(1+1/(3^s-1)). - Amiram Eldar, Dec 17 2022