Creative numbers

 Published on Saturday, 16th December 2017, 01:00 pm; Solved by 447;
Difficulty rating: 40%

Problem 616

Alice plays the following game, she starts with a list of integers $L$ and on each step she can either:

  • remove two elements $a$ and $b$ from $L$ and add $a^b$ to $L$
  • or conversely remove an element $c$ from $L$ that can be written as $a^b$, with $a$ and $b$ being two integers such that $a, b > 1$, and add both $a$ and $b$ to $L$

For example starting from the list $L=\{8\}$, Alice can remove $8$ and add $2$ and $3$ resulting in $L=\{2,3\}$ in a first step. Then she can obtain $L=\{9\}$ in a second step.

Note that the same integer is allowed to appear multiple times in the list.

An integer $n>1$ is said to be creative if for any integer $m>1$ Alice can obtain a list that contains $m$ starting from $L=\{n\}$.

Find the sum of all creative integers less than or equal to $10^{12}$.