%I
%S 0,0,0,0,0,0,0,1,0,2,0,1,1,1,1,2,0,2,1,1,2,2,0,1,1,2,3,2,1,1,1,2,2,1,
%T 0,1,2,3,3,2,2,2,2,2,2,3,2,1,0,2,3,3,3,1,2,2,4,2,1,0,3,1,3,4,1,3,4,2,
%U 4,3,2,1,2,3,4,2,3,3,0,3,5,2,4,0,1,3,2,3,4,4,3,2,5,5,3,0,5,4,6,3,3,1,3,2,3,5,3,0,4,2,3
%N Expansion of (Sum_{i>=1} x^prime(prime(i)))*(Sum_{j = p*q, p prime, q prime} x^j).
%C Number of ways of writing n as a sum of a prime with prime subscript (A006450) and a semiprime (A001358).
%C Every sufficiently large even number can be written as the sum of two primes, or a prime and a semiprime (Chen's theorem).
%C Conjecture: a(n) > 0 for all n > 527 (addition: only 18 positive integers cannot be represented as a sum of a prime number with prime subscript and a semiprime).
%H Ilya Gutkovskiy, <a href="/A282355/a282355.pdf">Extended graphical example</a>
%F G.f.: (Sum_{i>=1} x^prime(prime(i)))*(Sum_{j = p*q, p prime, q prime} x^j).
%e a(9) = 2 because we have [6, 3] and [5, 4].
%t nmax = 110; CoefficientList[Series[Sum[x^Prime[Prime[k]], {k, 1, nmax}] Sum[Floor[PrimeOmega[k]/2] Floor[2/PrimeOmega[k]] x^k, {k, 2, nmax}], {x, 0, nmax}], x]
%Y Cf. A001358, A006450, A100949.
%K nonn
%O 0,10
%A _Ilya Gutkovskiy_, Feb 13 2017
