In response to a comment posted under Powers of $2$ and the products of initial odd primes , I shall raise some questions about quantities near $O_n= P_{n+1}/2$, the product of the first $n$ odd primes. (Note the change in meaning of $O_n$; $P_n$ is the $n$th primorial.)

Before going into the questions, note that for integers $d$ of absolute value less than the largest prime factor of $O_n$ and coprime to $O_n$, such integers d have only 2 as a prime factor and $O_n + d$ has all its prime factors greater than the $n+1$st prime, unless $d$ has absolute value 1, in which case 2 is also a prime factor. Let this set of integers be called $C_n$.

(From the other question) How large a power of 2 divides $O_n^2 - 1$? As seen elsewhere, this quantity is not a power of 2 when $n>1$. More interestingly, the greater the power of 2 which divides this quantity, the further away one can demonstrate easily $O_n$ is from a power of 2.

Let $f(n)$ be that integer $e$ such that $2^e$ precisely divides $O_n^2 - 1$. So $f(1)=3, f(2)=5, f(3)=4.$ Is $f$ unbounded as a function of $n$? How fast does it grow? Is $f$ the sum of two nonconstant integer valued functions, one of them periodic and the other monotonic?

How many primes are in $C_n$?

Form $Q_n$, the product of the integers in $C_n$. This product is a multiple of 8, so is not squarefree. How far is it from being squarefree? In particular, is $Q_n$ the multiple of the square of any odd prime?

Given recent work on prime gaps (Zhang et al.), can we hope for a Generalized Dirichlet theorem? Specifically, translate the constellation $C_n$ by multiples of $O_n$: are there infinitely many translates which contain two or more primes in that translate?

I hope this makes my comment regarding related questions appear less vague and less broad.

admissible-set" tag? It's new to me, and I am not able to find anything about it (my Internet skills are weak). $\endgroup$6more comments