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Problem 632

Square prime factors

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.

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

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

平方质因数

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