## Divisor Nim

### Problem 509

Anton and Bertrand love to play three pile Nim.

However, after a lot of games of Nim they got bored and changed the rules somewhat.

They may only take a number of stones from a pile that is a proper divisor of the number of stones present in the pile.

E.g. if a pile at a certain moment contains 24 stones they may take only 1,2,3,4,6,8 or 12 stones from that pile.

So if a pile contains one stone they can't take the last stone from it as 1 isn't a proper divisor of 1.

The first player that can't make a valid move loses the game.

Of course both Anton and Bertrand play optimally.

The triple (`a`,`b`,`c`) indicates the number of stones in the three piles.

Let `S`(`n`) be the number of winning positions for the next player for 1 ≤ `a`, `b`, `c` ≤ `n`.`S`(10) = 692 and `S`(100) = 735494.

Find `S`(123456787654321) modulo 1234567890.