Number of ways to write 1 as sum of unit fractions

Question

For an integer $n \in \mathbb{N}$, let $f(n)$ be the number of ways to write 1 as a sum of exactly $n$ unit fractions. For example:

1. $f(1) = 1$ since there is only one way to write 1 as a sum of a single element.
2. $f(2) = 1$, since $1 = \frac{1}{2} + \frac{1}{2}$, and we can't sum up exactly two unit fraction to get 1.
3. (Not sure about this one actually) $f(3) = 3$, since $1 = \frac{1}{3} + \frac{1}{3} + \frac{1}{3}$, $1 = \frac{1}{2} + \frac{1}{4} + \frac{1}{4}$ and $\frac{1}{2} + \frac{1}{3} + \frac{1}{6}$ (perhaps there are more ways that I didn't think about).

Is there a way to estimate $f(n)$?

Answer 1

To add a little bit to BillyJoe's A002966 citation, there is not much known about $f(n)$. Only 8 values have been computed, and the values grow very quickly. Your $f(3) = 3$ is correct. Then it's

n	4	5	6	7	8
f(n)	14	147	3462	294,314	159,330,691

In Matthew Crawford's master's thesis linked in the OEIS entry, he makes two estimates for $f(9)$ (which he calls EFO(9)), $8.5 \times 10^{12}$ and $3.0 \times 10^{12}$.