Number of solutions to 4/prime(n) = 1/x + 1/y + 1/z for integers x, y, z with 0 < x < y < z.

0, 1, 2, 5, 7, 4, 4, 9, 19, 7, 17, 9, 7, 12, 32, 13, 25, 11, 15, 38, 7, 35, 25, 10, 8, 16, 25, 23, 15, 13, 31, 30, 17, 34, 18, 29, 24, 22, 63, 26, 45, 17, 65, 6, 23, 40, 28, 56, 35, 20, 19, 104, 8, 49, 19, 69, 28, 46, 31, 17, 31, 34, 38, 77, 16, 34, 36, 21, 37

1

The Erdos-Straus conjecture states that the equation has a solution for all n > 1. See S000342 and S000343 for the maximum values. See S000386 for the case in which equal terms are allowed.

T. D. Noe, Plot of 1000 terms

T. D. Noe, Table of 1000 terms

Kyle Bradford and Eugen Ionascu, A Geometric Consideration of the Erdős-Straus Conjecture, arXiv 1411.3403 (Nov 13 2014)

Eric W. Weisstein, MathWorld: Erdős-Straus Conjecture

(Mma) Needs["`Egypt`”]; Join[{0, 1}, Table[Length[EgyptianFraction[4/n, Method -> Lexicographic, MaxTerms -> 3, MinTerms -> 3, Duplicates -> Disallow, OutputFormat -> Plain]], {n, Prime[Range[3, 50]]}]]

Cf. A073101, S000342, S000343.

nonn

T. D. Noe, Nov 14 2014