Primes at which maximums occur in the Erdos-Straus conjecture graph (see S000341).
2, 3, 5, 7, 11, 19, 23, 47, 71, 167, 191, 239, 359, 479, 719, 839, 1319, 1439, 2399, 2879, 3119, 3359, 5039, 6719
1
Observe that for each of these primes p, p+1 is a number with small factors. For example 167+1 = 2^3 * 3 * 7. See S000343 for the maximum values.
T. D. Noe, Plot of 24 terms
Eric W. Weisstein, MathWorld: Erdős-Straus Conjecture
(Mma) Needs["`Egypt`”]; nn = 1000; t = Join[{0, 1}, Table[Length[EgyptianFraction[4/n, Method -> Lexicographic, MaxTerms -> 3, MinTerms -> 3, Duplicates -> Disallow, OutputFormat -> Plain]], {n, Prime[Range[3, nn]]}]]; t2 = {{2, 0}}; Do[If[t[[n]] > t2[[-1, 2]], AppendTo[t2, {Prime[n], t[[n]]}]], {n, 2, nn}]; Transpose[t2][[1]]
nonn
T. D. Noe, Nov 14 2014