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.

Given the function $f(N)$ and its properties, we know that the least significant digit of $n$ is $3$, and $n \le N$, which means the number $n$ are the numbers like $3, 13, 23,\dots, N$ (the last digit is $3$).

Let’s construct a prime number list as $p = [3,13,23,\dots, N]$. Because $f(N)$ is the smallest positive integer that is not coprime to any positive integer $n \le N$, we can infer $f(N)$ should be the product of all prime numbers in the sequence $p$. Because the smallest composite number is $4$ and for number $n \le N$, when $N \le 3$, the smallest positive integer is the number itself $N$, and when $N > 3$, the smallest positive integer $f(N)$ is the product of all prime numbers $p = [3,13,23,\dots, N]$.

We need to find the smallest positive number $f(10^6)$. First, let’s construct a prime number list $p = [3,13,23,\dots, 10^6]$ whose last digit is $3$. Then $f(10^6)$ equals to the product of all prime numbers of the list $p$.

Because it’s difficult to directly compute $f(10^6)$ and the question asks for the natural logarithm of $f(10^6)$, we can leverage the properties of natural log to simplify this. From logarithms, we know:

$\ln(ab) = \ln(a) + \ln(b)$

Therefore, $\ln(f(10^6))$ can be calculated as the sum of the natural logarithm of each prime number in the list. This method significantly simplifies the computation, then the problem becomes finding the sum of the natural logarithm of each prime number whose last digit is $3$ and less than or equal to $10^6$.

After this massive calculation, make sure to round the result to six digits after the decimal point.

But the above solution that directly calculates is not feasible in reality due to heavy computation. There might need some advanced mathematics theory or computer code to help you solve this problem.

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