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Comparing the divisors sequence $\sigma_0(n)$ and the $A(n)$ sequence, we observe that:  1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 4, 6, 2, 8, 2, 6 \\  0, 0, 0, 2, 0, 1, 0, 13,2, 4, 0, 1, 0, 6, 4, 4, 0, 1, 0, 15,1, 1, 0, 2, 3, 4, 4, 1, 0, 2, ... \\  \\  In the first $30$ terms [not that many, this is speculative] \\