CF1734C.Removing Smallest Multiples

You are given a set $S$ , which contains the first $n$ positive integers: $1, 2, \ldots, n$ .

You can perform the following operation on $S$ any number of times (possibly zero):

• Choose a positive integer $k$ where $1 \le k \le n$ , such that there exists a multiple of $k$ in $S$ . Then, delete the smallest multiple of $k$ from $S$ . This operation requires a cost of $k$ .

You are given a set $T$ , which is a subset of $S$ . Find the minimum possible total cost of operations such that $S$ would be transformed into $T$ . We can show that such a transformation is always possible.

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

The first line contains a single positive integer $n$ ( $1 \le n \le 10^6$ ).

The second line of each test case contains a binary string of length $n$ , describing the set $T$ . The $i$ -th character of the string is '1' if and only if $i$ is an element of $T$ , and '0' otherwise.

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

For each test case, output one non-negative integer — the minimum possible total cost of operations such that $S$ would be transformed into $T$ .

In the first test case, we shall not perform any operations as $S$ is already equal to $T$ , which is the set $\{1, 2, 3, 4, 5, 6\}$ .

In the second test case, initially, $S = \{1, 2, 3, 4, 5, 6, 7\}$ , and $T = \{1, 2, 4, 7\}$ . We shall perform the following operations:

1. Choose $k=3$ , then delete $3$ from $S$ .
2. Choose $k=3$ , then delete $6$ from $S$ .
3. Choose $k=5$ , then delete $5$ from $S$ .

The total cost is $3+3+5 = 11$ . It can be shown that this is the smallest cost possible.

In the third test case, initially, $S = \{1, 2, 3, 4\}$ and $T = \{\}$ (empty set). We shall perform $4$ operations of $k=1$ to delete $1$ , $2$ , $3$ , and $4$ .

In the fourth test case, initially, $S = \{1, 2, 3, 4\}$ and $T = \{3\}$ . We shall perform two operations with $k=1$ to delete $1$ and $2$ , then perform one operation with $k=2$ to delete $4$ .