# Representation of 1 as Sum of n Unit Fractions

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## Contents

## Theorem

Let $U \left({n}\right)$ denote the number of different ways of representing $1$ as the sum of $n$ unit fractions.

Then for various $n$, $U \left({n}\right)$ is given by the following table:

$n$ $U \left({n}\right)$ $1$ $1$ $2$ $1$ $3$ $3$ $4$ $14$ $5$ $147$ $6$ $3462$

This sequence is A002966 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

## Proof

Trivially:

- $1 = \dfrac 1 1$

and it follows that: $U \left({1}\right) = 1$

Also trivially:

- $1 = \dfrac 1 2 + \dfrac 1 2$

and it follows that: $U \left({2}\right) = 1$

From Sum of 3 Unit Fractions that equals 1:

- $U \left({3}\right) = 3$

From Sum of 4 Unit Fractions that equals 1:

- $U \left({4}\right) = 14$

From Sum of 5 Unit Fractions that equals 1:

- $U \left({5}\right) = 147$

## Historical Note

According to 1997: David Wells: *Curious and Interesting Numbers* (2nd ed.), this result is attributed to David Breyer Singmaster.

## Sources

- 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $147$