A15810. GCD Master (hard version)

This is the hard version of the problem. The only difference between the two versions is the constraint on $m$ . You can make hacks only if both versions of the problem are solved.

You are given an array $a$ of length $n$ and two integers $m$ and $k$ . Each element in $a$ satisfies $1\le a_i \le m$ .

In one operation, you choose two indices $i$ and $j$ such that $1 \le i < j \le |a|$ , then append $\gcd(a_i,a_j)$ to the back of the array and delete $a_i$ and $a_j$ from the array. Note that the length of the array decreases by one after this operation.

Find the maximum possible sum of the array after performing exactly $k$ operations.

The first line contains a single integer $t$ ( $1\le t\le 10^5$ ) — the number of test cases. The description of test cases follows.

The first line of each test case contains three integers $n$ , $m$ and $k$ ( $2 \le n \le 10^6$ ; $1\le m \le 9\cdot 10^{18}$ ; $1 \le k \le n-1$ ).

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

It is guaranteed that the sum of $n$ over all test cases does not exceed $10^6$ .

For each test case, output the maximum possible sum of the array after performing $k$ operations optimally.

In the first test case, the best way is to choose $i=1$ , $j=3$ in the first operation. The final sequence is $[7,4]$ .