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Problem 838

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Not Coprime

Let $f(N)$ be the smallest positive integer that is not coprime to any positive integer $n \le N$ 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.

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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。

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