A284943 Expansion of Sum_{p prime, k>=1} x^(p^k)*(1 - x)^2/(1 - 2*x)^2.

0, 1, 3, 8, 20, 47, 110, 251, 564, 1251, 2750, 5994, 12978, 27934, 59825, 127565, 270959, 573575, 1210466, 2547562, 5348385, 11203292, 23419629, 48865346, 101782870, 211670094, 439548898, 911515214, 1887865266, 3905400206, 8070139762, 16658958223, 34355273843

Offset: 1

Author

Ilya Gutkovskiy, Apr 06 2017

Keywords

• nonn

Comments

Total number of prime power parts (1 excluded) in all compositions (ordered partitions) of n.

Examples

			a(5) = 20 because we have [5], [4, 1], [3, 2], [3, 1, 1], [2, 3], [2, 2, 1], [2, 1, 2], [2, 1, 1, 1], [1, 4], [1, 3, 1], [1, 2, 2], [1, 2, 1, 1], [1, 1, 3], [1, 1, 2, 1], [1, 1, 1, 2], [1, 1, 1, 1, 1] and 1 + 1 + 2 + 1 + 2 + 2 + 2 + 1 + 1 + 1 + 2 + 1 + 1 + 1 + 1 + 0 = 20.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..3313
• Index entries for sequences related to compositions

Crossrefs

Cf. A011782, A059570, A073335, A097941, A097979, A102291, A246655, A284942.

Programs

Maple

b:= proc(n) option remember; nops(ifactors(n)[2])=1 end:
a:= proc(n) option remember; `if`(n=0, 0, add(a(n-j)+
      `if`(b(j), ceil(2^(n-j-1)), 0), j=1..n))
    end:
seq(a(n), n=1..33);  # Alois P. Heinz, Aug 07 2019

Mathematica

nmax = 33; Rest[CoefficientList[Series[Sum[Floor[1/PrimeNu[k]] x^k (1 - x)^2/(1 - 2 x)^2, {k, 2, nmax}], {x, 0, nmax}], x]]

PARI

x='x+O('x^34); concat([0], Vec(sum(k=2, 34, (1\omega(k))*x^k*(1 - x)^2/(1 - 2*x)^2))) \\ Indranil Ghosh, Apr 06 2017

Formula

G.f.: Sum_{p prime, k>=1} x^(p^k)*(1 - x)^2/(1 - 2*x)^2.