Question 13
Determine all positive integers $m$ with the following property:
The sequence $a_{0}, a_{1}, a_{2}, \ldots$ with $a_{0}=\frac{2m+1}{2}$ and $a_{k+1}=a_{k}\left\lfloor a_{k}\right\rfloor$ for $k=0,1,2, \ldots$ contains at least one integer.
Note: $\lfloor x\rfloor$ denotes the floor of $x$ (the greatest integer not exceeding $x$).
The sequence $a_{0}, a_{1}, a_{2}, \ldots$ with $a_{0}=\frac{2m+1}{2}$ and $a_{k+1}=a_{k}\left\lfloor a_{k}\right\rfloor$ for $k=0,1,2, \ldots$ contains at least one integer.
Note: $\lfloor x\rfloor$ denotes the floor of $x$ (the greatest integer not exceeding $x$).