%I #11 Feb 10 2017 20:18:27
%S 0,0,1,2,3,4,5,4,5,6,7,6,8,6,7,6,7,6,9,6,10,8,7,4,10,6,9,8,10,6,12,6,
%T 13,10,13,8,14,4,11,8,12,6,12,6,12,10,11,4,16,6,15,8,12,4,17,6,14,8,
%U 11,4,16,6,13,8,13,6,18,4,16,10,14,4,20,6,15,12,14,6,18,4,18,8,13,8,22,6,17,8,14,6,24,8,16,6,13,4
%N Expansion of (x + Sum_{p prime, k>=1} x^(p^k))^2.
%C Number of ways to write n as an ordered sum of two prime powers (1 included).
%H Robert Israel, <a href="/A282062/b282062.txt">Table of n, a(n) for n = 0..10000</a>
%H Ilya Gutkovskiy, <a href="/A282062/a282062.pdf">Extended graphical example</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimePower.html">Prime Power</a>
%F G.f.: (x + Sum_{p prime, k>=1} x^(p^k))^2.
%e a(8) = 5 because we have [7, 1], [5, 3], [4, 4], [3, 5] and [1, 7].
%p N:= 100: # to get a(0)..a(N)
%p P:= select(isprime, [$2..N]):
%p g:= x + add(add(x^(p^k),k=1..floor(log[p](N))),p=P):
%p S:= series(g^2,x,N+1):
%p seq(coeff(S,x,n),n=0..N); # _Robert Israel_, Feb 10 2017
%t nmax = 95; CoefficientList[Series[(x + Sum[Floor[1/PrimeNu[k]] x^k, {k, 2, nmax}])^2, {x, 0, nmax}], x]
%Y Cf. A000961, A071330, A071331, A073610, A095840, A280242, A282064.
%K nonn,look
%O 0,4
%A _Ilya Gutkovskiy_, Feb 05 2017