Definition:Balanced Prime/Definition 2

Definition

Let $\paren {p_{n - 1}, p_n, p_{n + 1} }$ be a triplet of consecutive prime numbers.

$p_n$ is a balanced prime if and only if:

$\ds p_{n - 1} + d$

$=$

$\ds p_n$

$\ds p_{n - 1} + 2 d$

$=$

$\ds p_{n + 1}$

for some $d \in \Z$.

That is, if and only if $p_n$ is the middle term of a $3$-term arithmetic sequence in which the $1$st and last terms are the previous and following primes.