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Denoting by      S ( N ) S(N)       the number of natural numbers      n n       less than      N N       for which  \[    4 n = 1 x + 1 y + 1 z \frac {4} {n} = \frac {1} {x} + \frac {1} {y} + \frac {1} {z}     \]  has no solutions in positive integers, we show that      S ( N ) ≪  N / log 2 N S(N) \ll N/{\log ^2}N      .

A note on 4/𝑛=1/𝑥+1/𝑦+1/𝑧