Does there exist a Benny number? For positive integers $x$, let $S(x)$ denote the sum of the  digits of $x$, and let $L(x)$ denote the number of digits of $x$. It can be shown that there are infinitely many numbers that cannot be expressed as $x+S(x)$ or $x+L(x)$ or $x+S(x)+L(x)$ individually or any method of those three i.e. $x+S(x)$, $x+L(x)$, and $x+S(x)+L(x)$ 


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*[Edit note: the question this is based on, Natural numbers not expressible as $x+s(x)$ nor $x+s(x)+l(x)$,  does not include $x+L(x)$ among the allowed formation methods].


And now a Benny number or Naughty number is a natural number greater than one that cannot be expressed as $x+S(x)$, $x-S(x)$, $x+L(x)$, $x-L(x)$, $x+S(x)+L(x)$, $x+S(x)-L(x)$, $x-S(x)+L(x)$, nor $x-S(x)-L(x)$. 
I've verified that there are no Benny numbers up to $10^{20}$. My question is: Does there exist a Benny number?
 A: For all $n$, we either have $L(n)=L(n+L(n))$ or $L(n)=L(n-L(n))$ (or both).  If $L(n)=L(n+L(n))$, let $x=n+L(n)$.  Then 
$$x-L(x)=(n+L(n))-L(n+L(n))=(n+L(n))-L(n)=n$$
Likewise, if $L(n)=L(n-L(n))$, let $x=n-L(n)$, in which case
$$x+L(x)=(n-L(n))+L(n-L(n))=(n-L(n))+L(n)=n$$
A: I want to explore the claim that "there are infinitely many numbers that cannot be expressed as $x+S(x)$ or $x+L(x)$ or $x+S(x)+L(x)$"
It is plain that, for most numbers $n$,  we can choose $x=n-L(n)$,  and for a few numbers after each power of $10$ we can choose $x=n-(L(n)-1)$, and then $n=x+L(x)$. In fact the only numbers which cannot be produced in this way are $n=10^k+k = \{11,102,1003,10004,\ldots\}$.
(With regard to the actual opening question, note that these can all be produced by selecting $x=10^k$ in $x-S(x)+L(x)$,  so yet another demonstration of no Benny numbers).
For $k<20$ and odd, selecting $x=10^k+(k-1)/2$ will give $n=x+S(x)$. However in general, if we can find a number $x$ such that $x+S(x)=10^k$,  then $x+S(x)+L(x)=10^k+k$. Can we always find such a number? The answer is that we can find such an $x$ with exceptions for every $10$th or so value of $k$: $\{6,16,26,36,46,57,67,77,\ldots\}$. 
Usually, the pattern of numbers reached by $x+S(x)$ means that we will find either $x+S(x)=10^k$ or $x+S(x)=10^k+k$ for some $x<10^k$ for numbers up to $k=46$, . However for $n=10^{46}+46, x=n=10^{46}+36$ gives $n=x+S(x)$. Likewise for $k=77$ we also have $ x=n=10^{77}+65$ gives $n=x+S(x)$, and $k=136, 167, 257,327, 527, 597, 687,\ldots$ are similarly handled.
So if there are exceptions to the rule that all numbers can be reached by one of $x+L(x),x+S(x)$ or $x+S(x)+L(x)$, they are exceedingly rare. I have found none up to $10^{1000}$ for sure.
