## Square prime factors

Published on Saturday, 28th July 2018, 01:00 pm; Solved by 331;
Difficulty rating: 35%

### Problem 632

For an integer $n$, we define the square prime factors of $n$ to be the primes whose square divides $n$. For example, the square prime factors of $1500=2^2 \times 3 \times 5^3$ are $2$ and $5$.

Let $C_k(N)$ be the number of integers between $1$ and $N$ inclusive with exactly $k$ square prime factors. You are given some values of $C_k(N)$ in the table below.

$\begin{array}{|c|c|c|c|c|c|c|} \hline & k = 0 & k = 1 & k = 2 & k = 3 & k = 4 & k = 5 \\ \hline N=10 & 7 & 3 & 0 & 0 & 0 & 0 \\ \hline N=10^2 & 61 & 36 & 3 & 0 & 0 & 0 \\ \hline N=10^3 & 608 & 343 & 48 & 1 & 0 & 0 \\ \hline N=10^4 & 6083 & 3363 & 533 & 21 & 0 & 0 \\ \hline N=10^5 & 60794 & 33562 & 5345 & 297 & 2 & 0 \\ \hline N=10^6 & 607926 & 335438 & 53358 & 3218 & 60 & 0 \\ \hline N=10^7 & 6079291 & 3353956 & 533140 & 32777 & 834 & 2 \\ \hline N=10^8 & 60792694 & 33539196 & 5329747 & 329028 & 9257 & 78 \\ \hline \end{array}$

Find the product of all non-zero $C_k(10^{16})$. Give the result reduced modulo $1\,000\,000\,007$.