Problem 550

Divisor game

Two players are playing a game. There are k piles of stones. When it is his turn a player has to choose a pile and replace it by two piles of stones under the following two conditions:

Both new piles must have a number of stones more than one and less than the number of stones of the original pile.
The number of stones of each of the new piles must be a divisor of the number of stones of the original pile.

The first player unable to make a valid move loses.
Let f(n,k) be the number of winning positions for the first player, assuming perfect play, when the game is played with k piles each having between 2 and n stones (inclusively).f(10,5)=40085.

Find f(10⁷,10¹²).Give your answer modulo 987654321.

约数游戏

两位玩家正在玩一个游戏：一开始，一共有k堆石子；每位玩家在轮到自己时，根据以下两个条件，选择一个堆，并将其换成两堆石子：

新的两堆石子的数目必须大于1且小于原来那一堆石子的数目；
新的两堆石子的数目必须是原来那一堆石子的数目的约数。

首先无法进行操作的玩家落败。
如果游戏一开始有k个堆，每堆石子的数目在2至n之间（含），记所有可能的游戏初始局面中先手玩家必胜的局面数为f(n,k)。已知f(10,5)=40085。

求f(10⁷,10¹²)，将你的答案对987654321取余。