OFFSET
0,10
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).
LINKS
Ilya Gutkovskiy, Extended graphical example
FORMULA
G.f.: (Sum_{i>=1} x^prime(prime(i)))*(Sum_{j = p*q, p prime, q prime} x^j).
EXAMPLE
a(9) = 2 because we have [6, 3] and [5, 4].
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]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Feb 13 2017
STATUS
approved