$\text{Notations}$
Let $\pi(n)$ be the prime counting function.
Let denote $\alpha(n)$ the sum of the prime factors of $n$. (i.e. In other words, if $$n=p_1^{x_1}p_2^{x_2}...p_m^{x_m}$$ then $\alpha(n)=p_1+p_2+...+p_m$.)
(I changed the notation; It was pointed out in the comments that $\omega$ is another function and it was misleading)
$\text{Statement}$
Find whether the equation $\alpha(n)+1|n+1$ has infinitely many squarefree solutions. Moreover, show that one can find an infinitude of composite, non-squarefree solutions.
$\text{Progress}$
The original question was posted here and Peter found the following general solution for non-squarefree integers:
Claim : $n=2\cdot 3\cdot 5^{5m+4}$ is a solution for all $m\ge 1$
Proof : The sum of the prime factors is obviously $10$. Because of $5^5\equiv 1\mod 11$ , we have $n\equiv 2\cdot 3\cdot 5^4\equiv -1\mod > 11$ , hence $11\mid n+1$ Hence there are infinite many solutions.
So it has been proven that there are in fact infinitely many non-squarefree solutions (and the prime numbers are trivial solutions). The question about composite squarefree solutions still holds.