What are all the possible triplets of numbers $a$, $b$, $c$ such that $a+2=b$, $a+4=c$, and all $3$ are prime powers (where one must be a power of $3$)?
I'm aware of the cases for when they are below $15$ and the sequence of numbers in the range $23$ to $31$, as well as the triplets $(79, 81, 83)$ and $(239, 241, 243)$, but I'm under the inclination that no others exist. This is part of a stronger question in which I'm attempting to determine the quadruplets satisfying a similar condition, but it was simple to deduce at least one of the elements is a power of $3$, and for these quadruplets must be one of the middle two elements (except for the smallest sequences which I've already noted above).
This itself is part of a larger question where I'm attempting to determine the increasing behavior of the function $$f(n)=\text{lcm}[1, 2, \ldots, n]-\Lambda(n) $$ (where $\Lambda(n)$ denotes the Von Mangoldt function), which just so happens to be increasing around these prime powers; the maximum length of increase happens when sequences of prime powers with difference no more than $2$ occurs; the longest sequences are in the two sequences I've noted above, and shorter increases elsewhere.