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Problem 269


Problem 269


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 Pn as the polynomial whose coefficients are the digits of n.
For example, P5703(x) = 5x3 + 7x2 + 3.

We can see that:

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

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

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

What is Z(1016)?


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Pn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:

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

在不超过k的正整数中,有些数n满足多项式Pn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?