Are there any theorems guaranteeing that if you add $1$, the number of prime factors will be reduced?

Question

Write $\Omega(n)$ for the number of prime factors of $n$ counting multiplicity. It's clear that $\Omega(2^n) > \Omega(2^n + 1)$. I'd like to know if there's a generalization of this of the form: "If $\Omega(n)$ is sufficiently large relative to $n$, then $\Omega(n) > \Omega(n+1)$."

Answer 1

If $\Omega(n) \ge \log_2(n)$, then $n$ is a power of $2$...

Answer 2

your property is pretty similar to $n$ being a product of primorials, i.e. non-increasing exponents. Add $1$ and every prime factor is larger than all prime factors of $n$ itself, so sum of exponents is smaller.

In turn, Ramanujan's procedure for finding a sequence of $n$ with extremely large $\Omega(n)$ is going to be disappointing here, most likely just $n = 2^j.$ We could force a multiplicative function by taking $$ f(n) = e^{\Omega(n)} \; \; , $$ perhaps $$ g(n) = 2^{\Omega(n)} \; \; . $$ That is actually an interesting problem, construct the analog of the Superior Highly Composite Numbers for this $f(n).$ The outcome might still be fairly trivial but I cannot quite be sure.