Question 55
Alan plays a solitaire game in his computer. To begin, Alan chooses a positive integer $n$.
Then he chooses a positive odd divisor of $n$.
If the chosen divisor is $d=1$, the computer replaces $n$ with $n+1$.
If the chosen divisor is a number $d$ greater than 1, the computer replaces $n$ with $n/d$.
Alan continues the game by choosing a divisor $d$ of the new number.
For instance:
$$
20 \xrightarrow{\{d=1\}} 21 \xrightarrow{\{d=3\}} 7 \xrightarrow{\{d=1\}} 8 \xrightarrow{\{\dots\}}
$$
Can Alan choose an appropriate starting number $n$ and play in such a way that the computer never shows a power of 3?
Then he chooses a positive odd divisor of $n$.
If the chosen divisor is $d=1$, the computer replaces $n$ with $n+1$.
If the chosen divisor is a number $d$ greater than 1, the computer replaces $n$ with $n/d$.
Alan continues the game by choosing a divisor $d$ of the new number.
For instance:
$$
20 \xrightarrow{\{d=1\}} 21 \xrightarrow{\{d=3\}} 7 \xrightarrow{\{d=1\}} 8 \xrightarrow{\{\dots\}}
$$
Can Alan choose an appropriate starting number $n$ and play in such a way that the computer never shows a power of 3?