Still confused. Following the description https://en.wikipedia.org/wiki/Grimm%27s_conjecture#Weaker_version, we have $ \omega \left ( \prod_{x\leq k}(n+x) \right )\geq k$ , ($ \omega (n)$ is the number of distinct prime factors of $ n$ ) with $ [n+1,n+k]\cap \mathbb{S}$ , where $ \mathbb{S}$ – set of composite numbers.

My goal is to find an upper bound for $ \omega \left ( \prod_{x\leq k}(n+x) \right )$ , with $ [n+1,n+k]\cap \mathbb{N}$ , where $ \mathbb{N}$ – set of natural numbers. Such an upper bound is also true for the interval $ [n+1,n+k]\cap \mathbb{S}$ , because $ \mathbb{S}\cap \mathbb{N}$ . We have $ \omega \left ( \prod_{x\leq k}(n+x) \right )< \sum_{n}^{n+k}\omega (n)$ . Because this $ \omega \left ( \prod_{x\leq k}(n+x) \right )$ expression only counts unique prime numbers up to $ n+x$ , but this $ \sum_{n}^{n+k}\omega (n)$ expression repeats (not all, perhaps) such unique prime numbers. Now we can write $ \sum_{n}^{n+k}\omega (n)=\sum_{p\leq n+k}\left \lfloor \frac{n+k}{p} \right \rfloor-\sum_{p\leq n}\left \lfloor \frac{n}{p} \right \rfloor$ . So $ \omega \left ( \prod_{x\leq k}(n+x) \right )<\sum_{p\leq n+k}\left \lfloor \frac{n+k}{p} \right \rfloor-\sum_{p\leq n}\left \lfloor \frac{n}{p} \right \rfloor$

IMPORTANT: Now you can remove duplicate primes in this $ \sum_{n}^{n+k}\omega (n)$ expression up to $ k$ . For example: in the interval $ [n+1,n+k]\cap \mathbb{N}$ – $ \frac{k}{2}$ numbers that are divisible by two and and also do not forget to save one unique number two, so we have $ -\frac{k}{2} + 1$ .

So, we have $ \omega \left ( \prod_{x\leq k}(n+x) \right )<\sum_{p\leq n+k}\left \lfloor \frac{n+k}{p} \right \rfloor-\sum_{p\leq n}\left \lfloor \frac{n}{p} \right \rfloor – \sum_{p\leq k}\left \lfloor \frac{k}{p} \right \rfloor + \pi (k)$

From this we can conclude that the maximum possible $ k$ will be $ k=n^{\frac{1}{c}}$ ($ k=n^{0.36…}$ , which is better than $ k=n^{1/2-\epsilon}$ in https://old.renyi.hu/~p_erdos/1971-24.pdf) – where $ c> 2.7182…$ (https://en.wikipedia.org/wiki/E_(mathematical_constant)).

And finally $ \omega \left ( \prod_{x\leq k}(n+x) \right )< Ck$

At what step did I get confused?