Binary Egyptian Fractions Original Research Article

Let A*k(n) be the number of positive integers a coprime to n such that the equation a/n=1/m1+…+1/mk admits a solution in positive integers (m1, …, mk). We prove that the sum of A*2(n) over nless-than-or-equals, slantx is both much greater-thanx log3 x and also much less-thanx log3 x. For the corresponding sum where the aʹs are counted with multiplicity of the number of solutions we obtain the asymptotic formula. We also show that A*k(n)much less-thannαk+var epsilon where αk is defined recursively by α2=0 and αk=1−(1−αk−1)/(2+αk−1).