IBM Ponder This #170 · June 2012

Two players starts with the number N and play in turns.
In each turn the player chooses a prime power p^m>1, which divides N and updates N to be N/(p^m).
The player who sets N to be 1 - wins.
What are all the winning moves from the initial state of N=1,506,009,006,001,500,000,000 ?

Update 6/8: We are looking for all the moves the first player can make which, assuming he plays correctly, can guarantee him winning the game.

Solution

Best opened after a real attempt

To be added.