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Problem 995

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A Particular Pair of Polynomials

For each prime $p$ and each positive integer $n$ define two polynomials:

$$
\begin{aligned}
f_p(x) &= \sum_{i=0}^{p-1}x^i \\
g_n(x) &= 1+\sum_{d\mid n}x^d
\end{aligned}
$$

Let $S(p)$ be the smallest positive integer $s$ such that $f_p(x)$ divides $g_s(x)$. For example, $S(2)=1$ as $f_2(x)=g_1(x)$. Also $S(5)=8$ because $f_5(x)\cdot(x^4-x^3+1)=g_8(x)$.

Define $T(m)$ to be the product of $S(p)$ over all primes $p \lt m$.
You are given that $T(20)=1348422598656$ and $T(100)\approx 1.37451\text{e}123$.

Find $T(20\ 000)$, giving your answer in scientific notation rounded to five significant digits after the decimal point. Use a lowercase $\text{e}$ to separate the mantissa and the exponent.

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一对特殊多项式

对任意质数$p$和任意正整数$n$，定义以下两个多项式：

$$
\begin{aligned}
f_p(x) &= \sum_{i=0}^{p-1}x^i \\
g_n(x) &= 1+\sum_{d\mid n}x^d
\end{aligned}
$$

记$S(p)$为使得$f_p(x)$整除$g_s(x)$的最小正整数$s$。例如，$S(2)=1$，因为$f_2(x)=g_1(x)$；又如$S(5)=8$，因为$f_5(x)\cdot(x^4-x^3+1)=g_8(x)$。

定义$T(m)$为所有小于$m$的质数$p$所对应的$S(p)$的乘积。
已知$T(20)=1348422598656$，$T(100)\approx 1.37451\text{e}123$。

求$T(20\ 000)$，将你的答案用科学计数法表示，四舍五入保留小数点后五位有效数字，使用小写字母$\text{e}$分隔尾数和指数。

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