I came across this question today.

Given an initial integer n0 > 1, two players, A and B, choose integers n1, n2,
n3, . . . alternately according to the following rules:
Knowing n2k, A chooses any integer n2k+1 such that
n2k · n2k+1 · n2
2k:
Knowing n2k+1, B chooses any integer n2k+2 such that
n2k+1
n2k+2
is a prime raised to a positive integer power.
Player A wins the game by choosing the number 1990; player B wins by choosing
the number 1. For which n0 does:
(a) A have a winning strategy?
(b) B have a winning strategy?
(c) Neither player have a winning strategy?