Lately I realized that strong Goldbach conjecture could be "reduced" to show that every even composite number with more than two prime factors (not necessarily distinct) can be expressed as the sum of two prime numbers. All even numbers of two prime factors are the sum of an odd prime number and itself (excepting $4$, which is $2+2$); thus, Goldbach conjecture holds for them, and so comes almost trivially the reformulation noticed.
Thinking about it, I was wondering if it could be established some connection between the number of prime factors of some even composite number, and its expressability as the sum of prime numbers (not necessarily two). For two prime factors is so easy, so I wonder if it could be not so hard for other numbers of prime factors. However, I do not know if someone has already explored this connection, or even if it could be worthy.
Trying to establish a simple upper bound for this connection, it can be noticed that, excepting powers of two, any even composite integer $n =2p_1p_2...p_k$, where $2\leq p_1\leq p_2\leq...\leq p_k$ can be expressed as a sum of at most $\frac{n}{p_k}$ prime numbers. With this bound, Goldbach conjecture for even composite integers of two prime factors follows inmediately. Any ideas on how to improve this bound with elementary methods?
Any comment on this would be welcomed. Thanks in advance!