# Is no language with the non-primes property context-free?

A language $L$ is said to have the "no primes" property if:

1. For every prime $p$ there are no words $w$ in $L$ s.t. $|w|=p$.

2. For every non-prime $m$ there is at least one word $w\in L$ of length $|w|=m$.

Given a language $L$ that has the no primes property, must $L$ be non-context-free?

Intuitively I believe yes, since the languages of all primes is not context free. But how can I prove this for every language that has this property?

I thiught proving using the sequence of lengths of words from any such language $L$, but this sequence is bounded by (at most) 3 (if 1 is consideres non prime, otherwise bounded by 2).

I also thiught proving by using the pumping lemma, but since I have no information on $L$ I am not sure if its possible.

Is there another method to prove so? Or a counter example that dissproves this?