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407. Idempotents

If we calculate a² mod 6 for 0 ≤ a ≤ 5 we get: 0,1,4,3,4,1.

The largest value of a such that a² ≡ a mod 6 is 4.
Let's call M(n) the largest value of a < n such that a² ≡ a (mod n).
So M(6) = 4.

Find ∑ M(n) for 1 ≤ n ≤ 10⁷.