A few updates need to be made related to Silger's 2002 publication of Fibonacci's 1202 AD Liber Abaci, which points out that a first and a second subtraction were allowed, thereby explaining several hard to evaluate data points in the n/17 table, as had been suggested by Brown and myself.
The Akhmim Papyrus was written around 500 AD. It is a Coptic text that used two forms of the Hultsch-Bruins method, a first and second subtraction. The first method: 2/p - 1/A = (2A -p)/AP was cited be Fibonacci per his methods 4, 5 and 6, as listed at the end of his seventh chapter, as aptly translated by Sigler. Note that Fibonacci's methods 4 and 5 stressed unit fraction first partitions, the style followed by Ahmes, while a medieval method 6 allowed vulgar fractions, such as 3/8 raised to a multiple of 6 or 18/48, to be a first partition.
The second Hultsch-Bruins method appearing in the Akhmim Papyrus is a modified form. The modification was read by Sylvester in 1891 as only an early version of the greedy algorithm. The Coptic version, and the data provided by Fibonacci, clearly show a second subtaction, as Leonardo (Fibonacci) lists as his 7th method, of 7, was used for 4/17 and several other n/17 table entries (i.e 8/17 and 13/17), and that n-subtractions as a greedy algorithm would require, were not mentioned by Fibonacci, nor by the Coptic scribe.
As a summary, beginning with Wilbur Knorr's 1982 Historia Mathematica data, the following analysis is pertinent, and therefore the data has been updated to be consistent with the RMP and the Liber Abaci.