Are twin primes more likely to be centered about $2^{p_i}$ where $p_i$ are the ordered primes?
Let us first define:
$$ (p^p \pm 1)/(p \pm 1) = \prod_{j=1}^{n}q_j, $$
where
$$ q_j = 1 \bmod p $$
for every $j$. We now have:
$$q_j = m_jp + 1$$
$$[\prod_{j=1}^n(m_j)(p) + 1)][(p \pm 1)] \mp 1 = p^p,$$
a beautiful start to a [beautiful] mess. $p$ is an odd prime $m_j > 1$ since when $m_j = 1$ is such that:
$$ ((1)(p) + 1) \nmid (p^p \pm 1)/(p \pm 1). $$
What is the nature of $n$? What are the restrictions on the value of $n$? What values of $m_j$ and how many $m_j$'s are there such that:
$$ ((m_j)(p) + 1) || (p^p \pm 1)? $$
(at least one).
As an exmaple, let's examine the case $p = 7$. $(7^7 + 1)/(7 + 1) = 137,257$ has only two prime factors $29\times4733$. $(7^7 - 1)/(7 - 1) = 102,943$ has only two prime factors: $113\times 911$.
I find it difficult to believe this is novel, although I wish it were.