Representation of 1 as Sum of n Unit Fractions

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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