A282500 Expansion of 1/(1 - Sum_{k = i^j, i>=1, j>=2} x^k).

1, 1, 1, 1, 2, 3, 4, 5, 8, 13, 19, 26, 37, 55, 81, 116, 167, 244, 358, 520, 752, 1091, 1589, 2311, 3354, 4870, 7081, 10298, 14963, 21734, 31580, 45900, 66704, 96919, 140827, 204654, 297413, 432180, 627996, 912565, 1326117, 1927054, 2800260, 4069160, 5913116, 8592675, 12486402, 18144506, 26366614

Offset: 0

Author

Ilya Gutkovskiy, Feb 16 2017

Keywords

• nonn

Comments

Number of compositions (ordered partitions) into perfect powers (A001597).

Examples

			a(7) = 5 because we have  [4, 1, 1, 1], [1, 4, 1, 1], [1, 1, 4, 1], [1, 1, 1, 4] and [1, 1, 1, 1, 1, 1, 1].
		

Links

• Eric Weisstein's World of Mathematics, Perfect Powers
• Index entries for sequences related to compositions

Crossrefs

Cf. A001597, A078635, A112344.

Programs

Mathematica

nmax = 95; CoefficientList[Series[1/ (1 - x - Sum[Boole[GCD @@ FactorInteger[k][[All, 2]] > 1] x^k, {k, 2, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - Sum_{k = i^j, i>=1, j>=2} x^k).

a(n) ~ c / r^n, where r = 0.68816189979082638501485812136220175833447947220530020978433949588627... and c = 0.4267808681995359684192168334905096310027880655306734537865362460298... . - Vaclav Kotesovec, Feb 17 2017