A154402 -id:A154402 - cp's OEIS frontend

A154403 Number of ways to write n as a sum a_1 + ... + a_k where a_i >= 2 * a_{i-1} > 1.

1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 11, 11, 12, 14, 14, 15, 17, 18, 19, 22, 23, 24, 27, 28, 30, 33, 35, 37, 40, 42, 44, 48, 50, 53, 58, 60, 63, 68, 71, 74, 80, 84, 87, 93, 97, 101, 107, 112, 117, 124, 129, 135, 143, 148, 155, 164, 170, 177, 187, 194

Offset: 0

Vladeta Jovovic, Jan 08 2009

Cf. A000929, A154402.

G.f.: x+(1/(1+x))/Product_{k>0}(1-x^(2^k-1)).

More terms from R. J. Mathar, Jan 21 2009

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

Original entry on oeis.org

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

Ilya Gutkovskiy, Oct 19 2019

nonn

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

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

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}]

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.

cp's OEIS Frontend

A154403 Number of ways to write n as a sum a_1 + ... + a_k where a_i >= 2 * a_{i-1} > 1.

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

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