A finite sequence of natural numbers, whose sum equal its product: The following pattern yields a finite sequence of natural numbers, whose sum equal its product:


*

*$A_1=k$

*$A_2=2$

*$A_3,\dots,A_k=1$


A few examples:


*

*$A_n=2,2$

*$A_n=3,2,1$

*$A_n=4,2,1,1$

*$A_n=5,2,1,1,1$

*$A_n=6,2,1,1,1,1$


Is there any other such sequence of natural numbers, which does not follow the pattern above?
Thanks

Take any finite sequence of natural numbers $x_1,x_2,\ldots,x_n$ such that $x_i\ne1$ for all $i$. Then $\prod\limits_{i=1}^nx_i\ge\sum\limits_{i=1}^nx_i$ and we can add to our sequence $\prod\limits_{i=1}^nx_i-\sum\limits_{i=1}^nx_i$ ones in order to satisfy the sum being equal to the product.
Note that this accounts for all possible such (unordered) sequences, apart from the simplest possible one: $x_1=1$.
For example take $x_1=3,x_2=3$. Then we can make the sequence $3,3,1,1,1$.
