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0562 number theory 5th edition Round 6 p2

Source:

May 6, 2021
number theory5th edition

Problem Statement

We say that a positive integer nn is nice if 4n\frac{4}{n} cannot be written as 1x+1xy+1z\frac{1}{x}+\frac{1}{xy}+\frac{1}{z} for any positive integers x,y,zx, y, z. Let us denote by an a_n the number of nice numbers smaller than nn. Prove that the sequence nan\frac{n}{a_n} is not bounded.
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