A16074 | Asterism Stream
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题目描述
Bogocubic is playing a game with amenotiomoi. First, Bogocubic fixed an integer $n$ , and then he gave amenotiomoi an integer $x$ which is initially equal to $1$ .
In one move amenotiomoi performs one of the following operations with the same probability:
- increase $x$ by $1$ ;
- multiply $x$ by $2$ .
Bogocubic wants to find the expected number of moves amenotiomoi has to do to make $x$ greater than or equal to $n$ . Help him find this number modulo $998\,244\,353$ .
Formally, let $M = 998\,244\,353$ . It can be shown that the answer can be expressed as an irreducible fraction $\frac{p}{q}$ , where $p$ and $q$ are integers and $q \not \equiv 0 \pmod{M}$ . Output the integer equal to $p \cdot q^{-1} \bmod M$ . In other words, output such an integer $y$ that $0 \le y < M$ and $y \cdot q \equiv p \pmod{M}$ .
In one move amenotiomoi performs one of the following operations with the same probability:
- increase $x$ by $1$ ;
- multiply $x$ by $2$ .
Bogocubic wants to find the expected number of moves amenotiomoi has to do to make $x$ greater than or equal to $n$ . Help him find this number modulo $998\,244\,353$ .
Formally, let $M = 998\,244\,353$ . It can be shown that the answer can be expressed as an irreducible fraction $\frac{p}{q}$ , where $p$ and $q$ are integers and $q \not \equiv 0 \pmod{M}$ . Output the integer equal to $p \cdot q^{-1} \bmod M$ . In other words, output such an integer $y$ that $0 \le y < M$ and $y \cdot q \equiv p \pmod{M}$ .
输入格式
Each test contains multiple test cases. The first line contains the number of test cases $t$ ( $1 \le t \le 100$ ). The description of the test cases follows.
The only line of each test case contains one integer $n$ ( $1 \le n \le 10^{18}$ ).
The only line of each test case contains one integer $n$ ( $1 \le n \le 10^{18}$ ).
输出格式
For each test case, output a single integer — the expected number of moves modulo $998\,244\,353$ .
输入输出样例
输入 #1
7 1 4 8 15 998244353 296574916252563317 494288321850420024
输出 #1
0 499122179 717488133 900515847 93715054 44488799 520723508
In the first test case, $n\le x$ without any operations, so the answer is $0$ .
In the second test case, for $n = 4$ , here is the list of all possible sequences of operations and their probabilities:
- $1\stackrel{+1}{\longrightarrow}2\stackrel{+1}{\longrightarrow}3\stackrel{+1}{\longrightarrow}4$ , the probability is $\frac{1}{8}$ ;
- $1\stackrel{\times 2}{\longrightarrow}2\stackrel{+1}{\longrightarrow}3\stackrel{+1}{\longrightarrow}4$ , the probability is $\frac{1}{8}$ ;
- $1\stackrel{+1}{\longrightarrow}2\stackrel{+1}{\longrightarrow}3\stackrel{\times 2}{\longrightarrow}6$ , the probability is $\frac{1}{8}$ ;
- $1\stackrel{\times 2}{\longrightarrow}2\stackrel{+1}{\longrightarrow}3\stackrel{\times 2}{\longrightarrow}6$ , the probability is $\frac{1}{8}$ ;
- $1\stackrel{+1}{\longrightarrow}2\stackrel{\times 2}{\longrightarrow}4$ , the probability is $\frac{1}{4}$ ;
- $1\stackrel{\times 2}{\longrightarrow}2\stackrel{\times 2}{\longrightarrow}4$ , the probability is $\frac{1}{4}$ .
So the expected number of moves is $4 \cdot \left(3 \cdot \frac{1}{8}\right) + 2 \cdot \left(2 \cdot \frac{1}{4} \right) =\frac{5}{2} \equiv 499122179 \pmod{998244353}$ .
In the third test case, for $n = 8$ , the expected number of moves is $\frac{137}{32}\equiv 717488133\pmod{998244353}$ .
In the fourth test case, for $n = 15$ , the expected number of moves is $\frac{24977}{4096} \equiv 900515847 \pmod{998244353}$ .
In the second test case, for $n = 4$ , here is the list of all possible sequences of operations and their probabilities:
- $1\stackrel{+1}{\longrightarrow}2\stackrel{+1}{\longrightarrow}3\stackrel{+1}{\longrightarrow}4$ , the probability is $\frac{1}{8}$ ;
- $1\stackrel{\times 2}{\longrightarrow}2\stackrel{+1}{\longrightarrow}3\stackrel{+1}{\longrightarrow}4$ , the probability is $\frac{1}{8}$ ;
- $1\stackrel{+1}{\longrightarrow}2\stackrel{+1}{\longrightarrow}3\stackrel{\times 2}{\longrightarrow}6$ , the probability is $\frac{1}{8}$ ;
- $1\stackrel{\times 2}{\longrightarrow}2\stackrel{+1}{\longrightarrow}3\stackrel{\times 2}{\longrightarrow}6$ , the probability is $\frac{1}{8}$ ;
- $1\stackrel{+1}{\longrightarrow}2\stackrel{\times 2}{\longrightarrow}4$ , the probability is $\frac{1}{4}$ ;
- $1\stackrel{\times 2}{\longrightarrow}2\stackrel{\times 2}{\longrightarrow}4$ , the probability is $\frac{1}{4}$ .
So the expected number of moves is $4 \cdot \left(3 \cdot \frac{1}{8}\right) + 2 \cdot \left(2 \cdot \frac{1}{4} \right) =\frac{5}{2} \equiv 499122179 \pmod{998244353}$ .
In the third test case, for $n = 8$ , the expected number of moves is $\frac{137}{32}\equiv 717488133\pmod{998244353}$ .
In the fourth test case, for $n = 15$ , the expected number of moves is $\frac{24977}{4096} \equiv 900515847 \pmod{998244353}$ .
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评测结果：Accepted