For $n\in \Bbb{Z}^+$ define the statement "$n$ is $k$-social" to mean that $$ \prod_{i=1}^n p_i +1 \mbox{ has exactly } k \mbox{ prime factors} $$ where $p_i$ is the $i$-th prime. So for example $5$ is $1$-social while $7$ is $3$-social. Similarly define "$n$ is $k$-antisocial to mean that $$ \prod_{i=1}^n p_i -1 \mbox{ has exactly } k \mbox{ prime factors} $$ For completeness, call $n=1$ to be $0$-antisocial.
(There may well be another more standard terminology for what I am calling a $k$-social number, and/or the prime number it generates. But these are different than the Euclid number primes.)
I had been trying decide the question of whether there is a largest $1$-social number (and similarly a largest $1$-antisocial number), and if there is no largest, find the asymptotic density and distribution of $1$-social numbers, but have not been able to make any progress beyond hand-waving arguments based on the distribution of primes.
[As an aside, $25$ is both $1$-social and $1$-antisocial; given the rarity of twin primes, that seems surprising.]
But here is a question which may be easier:
Define a number to be "friendly" if it is $k$-social and $m$-antisocial with $k>m$; "hostile" if it is $k$-social and $m$-antisocial with $k<m$; and "neutral" otherwise. Prove that the asymptotic density of friendly numbers equals that of hostile numbers. (I suspect that both are $\frac12$, and that neutral numbers become rare as $n \to \infty$.)
Although the original intent has been to treat factors of the form $p^j$ as a single prime factor, feel free to treat those as $j$ prime factors if it makes the problem easier.
