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# Problem 633

## Square prime factors II

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. It can be shown that with growing $N$ the ratio $\frac{C_k(N)}{N}$ gets arbitrarily close to a constant $c_k^\infty$, as suggested by the table below.

$k=0$ $k=1$ $k=2$ $k=3$ $k=4$
$C_k(10)$ $7$ $3$ $0$ $0$ $0$
$C_k(10^2)$ $61$ $36$ $3$ $0$ $0$
$C_k(10^3)$ $608$ $343$ $48$ $1$ $0$
$C_k(10^4)$ $6083$ $3363$ $533$ $21$ $0$
$C_k(10^5)$ $60794$ $33562$ $5345$ $297$ $2$
$C_k(10^6)$ $607926$ $335438$ $53358$ $3218$ $60$
$C_k(10^7)$ $6079291$ $3353956$ $533140$ $32777$ $834$
$C_k(10^8)$ $60792694$ $33539196$ $5329747$ $329028$ $9257$
$C_k(10^9)$ $607927124$ $335389706$ $53294365$ $3291791$ $95821$