This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A284943 #15 Aug 07 2019 17:17:49
%S A284943 0,1,3,8,20,47,110,251,564,1251,2750,5994,12978,27934,59825,127565,
%T A284943 270959,573575,1210466,2547562,5348385,11203292,23419629,48865346,
%U A284943 101782870,211670094,439548898,911515214,1887865266,3905400206,8070139762,16658958223,34355273843
%N A284943 Expansion of Sum_{p prime, k>=1} x^(p^k)*(1 - x)^2/(1 - 2*x)^2.
%C A284943 Total number of prime power parts (1 excluded) in all compositions (ordered partitions) of n.
%H A284943 Alois P. Heinz, <a href="/A284943/b284943.txt">Table of n, a(n) for n = 1..3313</a>
%H A284943 <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%F A284943 G.f.: Sum_{p prime, k>=1} x^(p^k)*(1 - x)^2/(1 - 2*x)^2.
%e A284943 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.
%p A284943 b:= proc(n) option remember; nops(ifactors(n)[2])=1 end:
%p A284943 a:= proc(n) option remember; `if`(n=0, 0, add(a(n-j)+
%p A284943 `if`(b(j), ceil(2^(n-j-1)), 0), j=1..n))
%p A284943 end:
%p A284943 seq(a(n), n=1..33); # _Alois P. Heinz_, Aug 07 2019
%t A284943 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]]
%o A284943 (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
%Y A284943 Cf. A011782, A059570, A073335, A097941, A097979, A102291, A246655, A284942.
%K A284943 nonn
%O A284943 1,3
%A A284943 _Ilya Gutkovskiy_, Apr 06 2017