An approximation to the twin prime conjecture and the parity phenomenon

Chen proved nearly 50 years ago that there are infinitely many primes p for which p+2 is either a prime or has exactly two prime factors. However we still do not know whether there are infinitely many primes p such that p+2 has an odd number of prime factors. The reason is the famous parity phenomenon. In the present work we show among others a weaker form of the above mentioned approximation to the twin prime conjecture, where the difference two is replaced by some positive even number below 20, which we cannot specify, however.