[USACOFeb2021G]StoneGame

Bessie and Elsie are playing a game with N (1≤N≤10⁵) piles of stones, where the i-th pile has ai stones for each 1≤i≤N (1≤a_i≤10⁶). The two cows alternate turns, with Bessie going first.

First, Bessie chooses some positive integer s₁ and removes s₁ stones from some pile with at least s₁ stones.
Then Elsie chooses some positive integer s₂ such that s₁ divides s₂ and removes s₂ stones from some pile with at least s₂ stones.
Then Bessie chooses some positive integer s₃ such that s₂ divides s₃ and removes s₃ stones from some pile with at least s₃ stones and so on.
In general, s_i, the number of stones removed on turn i, must divide s_i+1.

The first cow who is unable to remove stones on her turn loses.

Compute the number of ways Bessie can remove stones on her first turn in order to guarantee a win (meaning that there exists a strategy such that Bessie wins regardless of what choices Elsie makes). Two ways of removing stones are considered to be different if they remove a different number of stones or they remove stones from different piles.

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