Problem 616

Problem 616

Creative numbers

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 $10^{12}$.



  • 从$L$中去除两个元素$a$和$b$,并将$a^b$加入$L$
  • 或者相反地,从$L$中去除一个可以写成$a^b$形式的元素$c$,其中$a$和$b$均为整数且$a, b > 1$,然后将$a$和$b$加入$L$