A New Approach to Finding Upper Bounds for the "Powers of Two" Problem
Preliminary Note, September 25 2022
Max A. Alekseyev

Summary: I've developed a program that performs an analysis--somewhat similar to what Matthew Bolan did manually for a couple of graphs in his memo--that applies to any graph. 

Given a graph, it either finds all symbolic solutions associated with it or establishes that there are none. It is quite computationally expensive, and the number of graphs to consider is not small, so new values do not come easily.

Documenting all the details will require a full-length document.

However some values come "for free". 
Here is an example for a(14): suppose we have established that a(13)=21 and know that
27 = A006855(14) >= a(14) >= A347301(14) = 24. 
Notice that if a(14) >= 25, then the minimum vertex degree in the corresponding graph is at least 4 (as removing a vertex of a smaller degree would result in a graph for n=13 with > 21 edges). However, if the minimum degree is 4, then the number of edges is at least 14*4/2 = 28, greater than the upper bound for a(14). This contradiction proves that a(14) = 24.