IBM Ponder This #333 · January 2026

The decimal notation string of the number $123$ can be split in various ways: To $12$ and $3$ , to $1$ and $23$, to $1$, $2$ and $3$, and to $123$ . For each way to split the number, we add up the obtained parts. For $12$ and $3$ , we obtain the number $15$; for $1$ and $23$, we obtain the number $24$; for $1$, $2$, $3$ we obtain $6$; and for $123$ , we obtain $123$ itself. We denote this by:

$$A_{123}=\{6,15,24,123\}$$

For a general natural number $n$ , we define $A_n$ similarly as the set of all natural numbers that can be obtained by splitting and adding the decimal notation of $n$ . For example:

$$A_{31658}=\{23,32,50,68,77,95,104,176,329,374,662,689,1661,3173,31658\}$$

Given a set $A$ of integers and an integer $n$ we use the standard notation $n\cdot A\triangleq \{n\cdot x|x\in A\}$. (where · is the usual multiplication of numbers).

Your goal: Find the sum of all the natural numbers $x$ such that there exists $1\le n\le {10}^6$ for which $x\in n\cdot A_n$ and $n\in A_x$.

A bonus "*" will be given for finding the sum of all the natural numbers $x$ such that there exists $1\le n\le {10}^7$ for which $x\in n\cdot A_n$ and $n\in A_x$.

Solution

Best opened after a real attempt

To be added.