Polynomials with at least one integer root

A root or zero of a polynomial P(x) is a solution to the equation P(x) = 0. Define P_n as the polynomial whose coefficients are the digits of n. For example, P₅₇₀₃(x) = 5x³ + 7x² + 3.

We can see that:

• P_n(0) is the last digit of n,
• P_n(1) is the sum of the digits of n,
• P_n(10) is n itself.

Define Z(k) as the number of positive integers, n, not exceeding k for which the polynomial P_n has at least one integer root.

It can be verified that Z(100 000) is 14696.

What is Z(10¹⁶)?

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