$\text{Notations}$
Let $\pi(n)$ be the prime countiong function.
Let denote $\alpha(n)$ the sum of the prime factors of $n$. 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)
$\text{Statement}$
Prove or disprove that there exist infinitely many composite positive integers $n$ such that $\alpha(n)+1|n+1$.
$\text{Important}$
I made a new thread in which the question is posted with some new conditions. I am now interested in the squarefree solutions of the above equation. New problem link: Conjecture on the sum of prime factors
Best Answer
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.