A326812 Expansion of Sum_{k>=1} (2^k - 1) * x^(2^k - 1) / (1 - x^(2^k - 1)).

1, 1, 4, 1, 1, 4, 8, 1, 4, 1, 1, 4, 1, 8, 19, 1, 1, 4, 1, 1, 11, 1, 1, 4, 1, 1, 4, 8, 1, 19, 32, 1, 4, 1, 8, 4, 1, 1, 4, 1, 1, 11, 1, 1, 19, 1, 1, 4, 8, 1, 4, 1, 1, 4, 1, 8, 4, 1, 1, 19, 1, 32, 74, 1, 1, 4, 1, 1, 4, 8, 1, 4, 1, 1, 19, 1, 8, 4, 1, 1, 4, 1, 1, 11, 1

Offset: 1

Author

Ilya Gutkovskiy, Oct 19 2019

Keywords

• nonn

Comments

Sum of divisors of n of the form 2^j - 1 for j >= 1.

Crossrefs

Cf. A000225, A000929, A036987, A038712, A079559, A154402, A161790 (positions of 1's).

Programs

Mathematica

nmax = 85; CoefficientList[Series[Sum[(2^k - 1) x^(2^k - 1)/(1 - x^(2^k - 1)), {k, 1, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x] // Rest
Table[Sum[Mod[CatalanNumber[d], 2] d, {d, Divisors[n]}], {n, 1, 85}]

Formula

L.g.f.: -log(Product_{n>=1} (1 - x^(2^n - 1))) = Sum_{n>=1} a(n) * x^n / n.

exp(Sum_{n>=1} a(n) * x^n / n) = g.f. for A000929.

exp(Sum_{n>=1} (-1)^(n + 1) * a(n) * x^n / n) = g.f. for A079559.

a(n) = Sum_{d|n} A036987(d) * d.