A14573. 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.

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 $10^5$ .

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

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 $a = 3 + 1 + 1 + 0 + 0 + 0 = 5$ .