Let


$$ n\ := \sum\_{k=0}^L\ a\_k\cdot10^k $$
where $\ L\ $ is a non-negative integer, and $\ a\_k\ $ are decimal digits, and $\ a\_L>0.$


Let $\ n\ $ be called ***mixed*** $\ \Leftarrow:\Rightarrow$


$$ n\ =\ \sum\_{k=0}^L a\_k\cdot\prod\_{k=0}^L a\_k $$


For instance, $\,\ n:=1\,\ $ and $\,\ n:=144\ $ are mixed.


**Problem 1:**   describe all mixed numbers.


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One may also consider mixed numbers $\,\ n:=\sum\_{k=0}^L\, a\_k\cdot\beta^k\,\ $
base $\ \beta,\ $ where $\ 1<\beta\in\mathbb Z.\ $


The only mixed number base $\ 2\ $ is $\ n:=1.$


In general,


$\quad$


$\qquad\qquad$ for arbitrary base $\ \beta\ $ there are only


$\qquad\qquad$ finitely many mixed numbers base $\ \beta.$


$\quad$


**Problem 2:**   describe all mixed numbers for arbitrary base $\ \beta$.




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**Remark:**   The case of odd base $\ \beta\ $ is distinctly different from the even case.




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Diophantine entertainment -- mixed natural numbers
<https://oeis.org/A038369> "Numbers $k$ such that $k$ = (product of digits of $k$) times (sum of digits of $k$)."


"$0, 1, 135, 144$. The list is complete. Proof: One shows that the number of digits is at most $84$ and then it is only necessary to consider numbers of the forms $2^i3^j7^k$ and $3^i5^j7^k$. - David W. Wilson, May 16 2003."


Many references to the literature are given at that page. In particular, Ezra Bussmann, S·P Numbers in bases other than 10, The Mathematical Gazette, 85(503)(2001) 245–248. <https://doi.org/10.2307/3622010> finds all the examples in all bases $b$, $2\le b\le12$. $1$ is an example in every base; the others are


Base $4$: $12$  

$5$: $341$  

$7$: $22; 242; 1254; 2343; 116655; 324236; 424644$  

$9$: $13; 281876; 724856; 7487248$  

$10$: $135; 144$  

$11$: $253; 419; 2189; 7634; 82974$  

$12$: $128; 173; 353$


