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.

	$k = 0$	$k = 1$	$k = 2$	$k = 3$	$k = 4$
$C_{k} (10)$	$7$	$3$	$0$	$0$	$0$
$C_{k} (10^{2})$	$61$	$36$	$3$	$0$	$0$
$C_{k} (10^{3})$	$608$	$343$	$48$	$1$	$0$
$C_{k} (10^{4})$	$6083$	$3363$	$533$	$21$	$0$
$C_{k} (10^{5})$	$60794$	$33562$	$5345$	$297$	$2$
$C_{k} (10^{6})$	$607926$	$335438$	$53358$	$3218$	$60$
$C_{k} (10^{7})$	$6079291$	$3353956$	$533140$	$32777$	$834$
$C_{k} (10^{8})$	$60792694$	$33539196$	$5329747$	$329028$	$9257$
$C_{k} (10^{9})$	$607927124$	$335389706$	$53294365$	$3291791$	$95821$
$c_{k}^{\infty}$	$\frac{6}{π^{2}}$	$3.3539 \times 10^{- 1}$	$5.3293 \times 10^{- 2}$	$3.2921 \times 10^{- 3}$	$9.7046\times 10^{-5}

Find $c_{7}^{\infty}$ . Give the result in scientific notation rounded to $5$ significant digits, using a $e$ to seperate mantissa and exponent. E.g., if the answer were $0.000123456789$ , then the answer format would be $1.2346 e - 4$ .

平方质因数II

对于整数 $n$ ，称平方后能整除 $n$ 的质数为 $n$ 的平方质因数。例如， $1500 = 2^{2} \times 3 \times 5^{3}$ 的平方质因数包括 $2$ 和 $5$ 。

记 $C_{k} (N)$ 为 $1$ 与 $N$ 之间（含 $1$ 和 $N$ ）恰好有 $k$ 个平方质因数的整数之和。可以发现，随着 $N$ 不断增大，比值 $\frac{C_{k} (N)}{N}$ 逐渐趋向于一个常数 $c_{k}^{\infty}$ ，如下表所示：

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

求 $c_{7}^{\infty}$ ，并将结果用科学计数法表示，保留5位有效数字，尾数和指数间用 $e$ 分隔。例如，如果结果是 $0.000123456789$ ，则应当回答 $1.2346 e - 4$ 。