1734C · Removing Smallest Multiples

Problem: 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.

Input Format: 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 $10^6$.

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

Note: 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 $4$.