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

0, 2, 2, 6, 8, 4, 4, 10, 20, 7, 18, 9, 7, 13, 33, 13, 26, 11, 16, 39, 7, 36, 26, 10, 8, 16, 26, 24, 15, 13, 32, 31, 17, 35, 18, 30, 24, 23, 64, 26, 46, 17, 66, 6, 23, 41, 29, 57, 36, 20, 19, 105, 8, 50, 19, 70, 28, 47, 31, 17, 32, 34, 39, 78, 16, 34, 37, 21, 38

1

Here equality is allowed. See S000341 where the terms 1/x, 1/y, and 1/z are different. The only differences occur at primes of the form 4k+3, where this sequence is 1 greater than the corresponding term in S000341.

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, 2}, Table[Length[EgyptianFraction[4/n, Method -> Lexicographic, MaxTerms -> 3, MinTerms -> 3, Duplicates -> Allow, OutputFormat -> Plain]], {n, Prime[Range[3, 50]]}]]

Cf. S000341 (three different terms).

nonn

T. D. Noe, Nov 25 2014