$n$ numbers with their sum and product equal to each other.

Question

I was trying to do this:

For any integer $n\ge2$, find $n$ numbers $a_1,\ a_2,\ a_3,\dots,a_n$ such that $$a_1+a_2+a_3+\dots+a_n=a_1a_2a_3\dots a_n$$

For $n=2$, we have $a_1=a_2=2$. (As $2+2=2\times2)$

For $n=3$, we have $a_1=1,\ a_2=2,\ a_3=3.$ ($1+2+3=1\times2\times3$)

Do there exist any such numbers for higher values of $n$ (i.e $n\ge4$). If there are give such examples. If there don't exist such numbers for $n\ge4$, there must be a proof from elementary number theory for this. What is that proof?

Answer 1

Have a look at http://www-users.mat.umk.pl/~anow/ps-dvi/si-krl-a.pdf.

From therein, further examples are $1,1,2,4$ and $1,1,2,2,2$. It is also shown that for every $n$ the number of such sequences is positive (just take $1,1,\dots,1,2,n$) and finite.

See below for the number of such sequences $a(n)$ for $1\le n\le 100$.