CF1603C.Extreme Extension

For an array $b$ of $n$ integers, the extreme value of this array is the minimum number of times (possibly, zero) the following operation has to be performed to make $b$ non-decreasing:

• Select an index $i$ such that $1 \le i \le |b|$, where $|b|$ is the current length of $b$.
• Replace $b_i$ with two elements $x$ and $y$ such that $x$ and $y$ both are positive integers and $x + y = b_i$.
• This way, the array $b$ changes and the next operation is performed on this modified array.

For example, if $b = [2, 4, 3]$ and index $2$ gets selected, then the possible arrays after this operation are $[2, \underline{1}, \underline{3}, 3]$, $[2, \underline{2}, \underline{2}, 3]$, or $[2, \underline{3}, \underline{1}, 3]$. And consequently, for this array, this single operation is enough to make it non-decreasing: $[2, 4, 3] \rightarrow [2, \underline{2}, \underline{2}, 3]$.

It's easy to see that every array of positive integers can be made non-decreasing this way.

YouKn0wWho has an array $a$ of $n$ integers. Help him find the sum of extreme values of all nonempty subarrays of $a$ modulo $998\,244\,353$. If a subarray appears in $a$ multiple times, its extreme value should be counted the number of times it appears.

An array $d$ is a subarray of an array $c$ if $d$ can be obtained from $c$ by deletion of several (possibly, zero or all) elements from the beginning and several (possibly, zero or all) elements from the end.

对于一个包含 $n$ 个整数的数组 $b$，该数组的极值（extreme value）定义为：使其变为非递减数组所需执行以下操作的最少次数（可以为零次）：

• 选择一个下标 $i$，满足 $1 \le i \le |b|$，其中 $|b|$ 表示当前数组 $b$ 的长度；
• 将 $b_i$ 替换为两个正整数 $x$ 和 $y$，使得 $x + y = b_i$；
• 经此操作后，数组 $b$ 发生变化，后续操作在该修改后的数组上进行。

例如，若 $b = [2, 4, 3]$ 且选中下标 $2$，则此次操作后可能得到的数组为 $[2, \underline{1}, \underline{3}, 3]$、$[2, \underline{2}, \underline{2}, 3]$ 或 $[2, \underline{3}, \underline{1}, 3]$。因此，对这个数组，仅需一次操作即可使其非递减：$[2, 4, 3] \rightarrow [2, \underline{2}, \underline{2}, 3]$。

显然，任意由正整数组成的数组均可通过上述方式变为非递减数组。

YouKn0wWho 拥有一个长度为 $n$ 的整数数组 $a$。请帮助他计算 $a$ 的所有非空子数组的极值之和，并对 $998\,244\,353$ 取模。若某个子数组在 $a$ 中多次出现，则其极值应按其出现次数累加。

数组 $d$ 是数组 $c$ 的子数组（subarray），当且仅当 $d$ 可通过从 $c$ 的开头删除若干个（可以为零个或全部）元素、并从 $c$ 的末尾删除若干个（可以为零个或全部）元素而得到。

The first line contains a single integer $t$ ($1 \le t \le 10\,000$) — the number of test cases.

The first line of each test case contains a single integer $n$ ($1 \le n \le 10^5$).

The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^5$).

It is guaranteed that the sum of $n$ over all test cases doesn't exceed $10^5$.

第一行包含一个整数 $t$（$1 \le t \le 10\,000$）—— 表示测试用例的数量。

每个测试用例的第一行包含一个整数 $n$（$1 \le n \le 10^5$）。

每个测试用例的第二行包含 $n$ 个整数 $a_1, a_2, \ldots, a_n$（$1 \le a_i \le 10^5$）。

保证所有测试用例的 $n$ 之和不超过 $10^5$。

For each test case, print a single integer — the sum of extreme values of all subarrays of $a$ modulo $998\,244\,353$.

对于每个测试用例，输出一个整数——数组 $a$ 的所有子数组的极值之和对 $998\,244\,353$ 取模的结果。

Let $f(l, r)$ denote the extreme value of $[a_l, a_{l+1}, \ldots, a_r]$.

In the first test case,

• $f(1, 3) = 3$, because YouKn0wWho can perform the following operations on the subarray $[5, 4, 3]$ (the newly inserted elements are underlined):

  $[5, 4, 3] \rightarrow [\underline{3}, \underline{2}, 4, 3] \rightarrow [3, 2, \underline{2}, \underline{2}, 3] \rightarrow [\underline{1}, \underline{2}, 2, 2, 2, 3]$;

• $f(1, 2) = 1$, because $[5, 4] \rightarrow [\underline{2}, \underline{3}, 4]$;

• $f(2, 3) = 1$, because $[4, 3] \rightarrow [\underline{1}, \underline{3}, 3]$;

• $f(1, 1) = f(2, 2) = f(3, 3) = 0$, because they are already non-decreasing.

So the total sum of extreme values of all subarrays of $a = 3 + 1 + 1 + 0 + 0 + 0 = 5$.

令 $f(l, r)$ 表示子数组 $[a_l, a_{l+1}, \ldots, a_r]$ 的极值。

在第一个测试用例中，

• $f(1, 3) = 3$，因为 YouKn0wWho 可对子数组 $[5, 4, 3]$（新插入的元素用下划线标出）执行如下操作：

  $[5, 4, 3] \rightarrow [\underline{3}, \underline{2}, 4, 3] \rightarrow [3, 2, \underline{2}, \underline{2}, 3] \rightarrow [\underline{1}, \underline{2}, 2, 2, 2, 3]$；

• $f(1, 2) = 1$，因为 $[5, 4] \rightarrow [\underline{2}, \underline{3}, 4]$；

• $f(2, 3) = 1$，因为 $[4, 3] \rightarrow [\underline{1}, \underline{3}, 3]$；

• $f(1, 1) = f(2, 2) = f(3, 3) = 0$，因为这些单元素子数组本身已为非递减序列。

因此，数组 $a$ 的所有子数组的极值之和为 $3 + 1 + 1 + 0 + 0 + 0 = 5$。