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No.1277 Series within a Series

レベル : / 実行時間制限 : 1ケース 2.000秒 / メモリ制限 : 512 MB / 標準ジャッジ問題
タグ : （AC するまで非表示） / 解いたユーザー数 15
作問者 : 👑 qand / テスター : 57tggx
お気に入りにしたユーザー ProblemId : 5402 / 自分の提出
問題文最終更新日: 2022-11-13 12:30:00
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yukicoder contest 272 (Weird math contest) (順位表) の他の問題:

Note

This problem relies on the assumption that $\zeta(2),\zeta(3),\ldots$ are linearly independent over $\mathbb{Z}$, but this was an open problem. The problem has therefore been moved to the unproven / flawed problems category. We sincerely apologize.

Problem Statement

Kamitori was given the following problem by PCT.

$S_n=\displaystyle\sum_{\displaystyle a_1,\ldots,a_n>0} \frac1{\displaystyle\prod_{k=1}^n a_k\left(\left(\sum_{k=1}^n a_k\right)-1 \right)}$

とおきます。すると、

$\displaystyle S_n=\sum_{i=1}^m b_i\zeta(c_i)$

を満たす長さ $m$ の整数列 $b,c$ と正整数 $m$ が存在します。

このような $b,c,m$ のうち、$m$ が最小となるような $b,c,m$ を求めてください。
However, Kamitori cannot solve this problem. Please solve it in Kamitori's place.

It is guaranteed that this series converges within the specified range.

Input

$n$

  • All inputs are integers.
  • $1 < n \le 10^{7}$

    • Output

    On the first line, output the value of $m$.

    On the second line, output the product of all elements of $b$, modulo $10^9+7$.

    On the third line, output the product of all elements of $c$, modulo $10^9+7$.

    It is guaranteed that these values are uniquely determined.

    Here, the remainder of an integer $x$ divided by $10^9+7$ means the smallest non-negative integer $p$ satisfying $x ≡ p (mod 10^9+7)$.

    Print a newline at the end.

    Sample

    Sample 1
    Input
    2
    Output
    1
2
2

    We have $\displaystyle\sum_{a_1,a_2>0} \frac1{a_1a_2(a_1+a_2-1)}=2\zeta(2)$.

    In this case, $m=1$, the product of the elements of $b$ is $2$, and the product of the elements of $c$ is $2$, so output as shown above.

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