A282355 Expansion of (Sum_{i>=1} x^prime(prime(i)))*(Sum_{j = p*q, p prime, q prime} x^j).

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, 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, 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

Offset: 0

Author

Ilya Gutkovskiy, Feb 13 2017

Keywords

• nonn

Comments

Number of ways of writing n as a sum of a prime with prime subscript (A006450) and a semiprime (A001358).

Every sufficiently large even number can be written as the sum of two primes, or a prime and a semiprime (Chen's theorem).

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).

Examples

			a(9) = 2 because we have [6, 3] and [5, 4].
		

Links

• Ilya Gutkovskiy, Extended graphical example

Crossrefs

Cf. A001358, A006450, A100949.

Programs

Mathematica

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]

Formula

G.f.: (Sum_{i>=1} x^prime(prime(i)))*(Sum_{j = p*q, p prime, q prime} x^j).