Problem 838

Not Coprime

Let $f (N)$ be the smallest positive integer that is not coprime to any positive integer $n \leq N$ whose least significant digit is $3$ .

For example $f (40)$ equals to $897 = 3 \cdot 13 \cdot 23$ since it is not coprime to any of $3, 13, 23, 33$ . By taking the natural logarithm (log to base $e$ ) we obtain $\ln f (40) = \ln 897 \approx 6.799056$ when rounded to six digits after the decimal point.

You are also given $\ln f (2800) \approx 715.019337$ .

Find $f (10^{6})$ . Enter its natural logarithm rounded to six digits after the decimal point.

不互质

记 $f (N)$ 为最小的、与所有小于等于 $N$ 且末位为 $3$ 的数都不互质的正整数。

例如， $f (40)$ 等于 $897 = 3 \cdot 13 \cdot 23$ ，因为它和 $3, 13, 23, 33$ 中的任意一个都不互质。对这个数取自然对数（以 $e$ 为底的对数）并保留六位小数，可得 $\ln f (40) = \ln 897 \approx 6.799056$ 。

已知 $\ln f (2800) \approx 715.019337$ 。

求 $f (10^{6})$ ，取其自然对数并保留六位小数作为答案。