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Problem 269

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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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有至少一个整数根的多项式

多项式P(x)的根或零点，是方程P(x) = 0的解。
定义P_n为系数等于n的各位数字的多项式。
例如，P₅₇₀₃(x) = 5x³ + 7x² + 3。

我们可以看出：

• P_n(0)是n的个位数字。
• P_n(1)是n的各位数字和。
• P_n(10)是n本身。

在不超过k的正整数中，有些数n满足多项式P_n至少有一个整数根，定义Z(k)为这类数的数目。

可以验证Z(100 000)为14696。

Z(10¹⁶)是多少？

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