310731: CF1877D. Effects of Anti Pimples

D. Effects of Anti Pimplestime limit per test2 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard output

Chaneka has an array $[a_1,a_2,\ldots,a_n]$. Initially, all elements are white. Chaneka will choose one or more different indices and colour the elements at those chosen indices black. Then, she will choose all white elements whose indices are multiples of the index of at least one black element and colour those elements green. After that, her score is the maximum value of $a_i$ out of all black and green elements.

There are $2^n-1$ ways for Chaneka to choose the black indices. Find the sum of scores for all possible ways Chaneka can choose the black indices. Since the answer can be very big, print the answer modulo $998\,244\,353$.

Input

The first line contains a single integer $n$ ($1 \leq n \leq 10^5$) — the size of array $a$.

The second line contains $n$ integers $a_1,a_2,a_3,\ldots,a_n$ ($0\leq a_i\leq10^5$).

Output

An integer representing the sum of scores for all possible ways Chaneka can choose the black indices, modulo $998\,244\,353$.

ExamplesInput

4
19 14 19 9

Output

265

Input

1
0

Output

0

Input

15
90000 9000 99000 900 90900 9900 99900 90 90090 9090 99090 990 90990 9990 99990

Output

266012571

Note

In the first example, below are the $15$ possible ways to choose the black indices:

• Index $1$ is black. Indices $2$, $3$, and $4$ are green. Maximum value among them is $19$.
• Index $2$ is black. Index $4$ is green. Maximum value among them is $14$.
• Index $3$ is black. Maximum value among them is $19$.
• Index $4$ is black. Maximum value among them is $9$.
• Indices $1$ and $2$ are black. Indices $3$ and $4$ are green. Maximum value among them is $19$.
• Indices $1$ and $3$ are black. Indices $2$ and $4$ are green. Maximum value among them is $19$.
• Indices $1$ and $4$ are black. Indices $2$ and $3$ are green. Maximum value among them is $19$.
• Indices $2$ and $3$ are black. Index $4$ is green. Maximum value among them is $19$.
• Indices $2$ and $4$ are black. Maximum value among them is $14$.
• Indices $3$ and $4$ are black. Maximum value among them is $19$.
• Indices $1$, $2$, and $3$ are black. Index $4$ is green. Maximum value among them is $19$.
• Indices $1$, $2$, and $4$ are black. Index $3$ is green. Maximum value among them is $19$.
• Indices $1$, $3$, and $4$ are black. Index $2$ is green. Maximum value among them is $19$.
• Indices $2$, $3$, and $4$ are black. Maximum value among them is $19$.
• Indices $1$, $2$, $3$, and $4$ are black. Maximum value among them is $19$.

The total sum is $19+14+19+9+19+19+19+19+14+19+19+19+19+19+19 = 265$.

题目大意：
Chaneka有一个数组[a_1, a_2, ..., a_n]。最初，所有元素都是白色的。Chaneka将选择一个或多个不同的索引，并将选定索引处的元素涂成黑色。然后，她会选择所有索引为至少一个黑色元素的倍数的白色元素，并将这些元素涂成绿色。在此之后，她的得分是所有黑色和绿色元素中a_i的最大值。

有2^n-1种方式供Chaneka选择黑色索引。计算Chaneka可以选择黑色索引的所有可能方式的得分总和。由于答案可能非常大，请以998,244,353为模输出答案。

输入数据格式：
第一行包含一个整数n（1≤n≤10^5）——数组a的大小。
第二行包含n个整数a_1, a_2, a_3, ..., a_n（0≤a_i≤10^5）。

输出数据格式：
一个整数，代表Chaneka可以选择黑色索引的所有可能方式的得分总和，模998,244,353。题目大意： Chaneka有一个数组[a_1, a_2, ..., a_n]。最初，所有元素都是白色的。Chaneka将选择一个或多个不同的索引，并将选定索引处的元素涂成黑色。然后，她会选择所有索引为至少一个黑色元素的倍数的白色元素，并将这些元素涂成绿色。在此之后，她的得分是所有黑色和绿色元素中a_i的最大值。 有2^n-1种方式供Chaneka选择黑色索引。计算Chaneka可以选择黑色索引的所有可能方式的得分总和。由于答案可能非常大，请以998,244,353为模输出答案。 输入数据格式： 第一行包含一个整数n（1≤n≤10^5）——数组a的大小。 第二行包含n个整数a_1, a_2, a_3, ..., a_n（0≤a_i≤10^5）。 输出数据格式： 一个整数，代表Chaneka可以选择黑色索引的所有可能方式的得分总和，模998,244,353。