Unit fractions

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A unit fraction is a fraction having a numerator of 1. For example, $\scriptstyle \frac{1}{7}\,$ is a unit fraction.

Problems involving unit fractions

There are various problems in number theory involving unit fractions.

Smallest k such that 1/k can be written as a sum of exactly 2 unit fractions in n ways

The smallest integer $\scriptstyle k \,$ such that $\scriptstyle \frac{1}{k} \,$ can be written as a sum of two unit fractions in exactly $\scriptstyle n \,$ ways is given by (Cf. A016017)

{1, 2, 4, 8, 6, 32, 64, 12, 256, 512, 24, 2048, 36, 30, 16384, 32768, 96, 72, 262144, 192, 1048576, 2097152, 60, 8388608, 216, 768, 67108864, 288, 1536, 536870912, ...}

Erdős-Straus conjecture

The Erdős-Straus conjecture says that

$\frac{4}{n} = \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \,$

always has solutions in integers.

Unit fractions

From OeisWiki

Contents

Problems involving unit fractions

Smallest k such that 1/k can be written as a sum of exactly 2 unit fractions in n ways

Erdős-Straus conjecture

See also

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