CF1744D.Divisibility by 2^n

You are given an array of positive integers $a_1, a_2, \ldots, a_n$.

Make the product of all the numbers in the array (that is, $a_1 \cdot a_2 \cdot \ldots \cdot a_n$) divisible by $2^n$.

You can perform the following operation as many times as you like:

• select an arbitrary index $i$ ($1 \leq i \leq n$) and replace the value $a_i$ with $a_i=a_i \cdot i$.

You cannot apply the operation repeatedly to a single index. In other words, all selected values of $i$ must be different.

Find the smallest number of operations you need to perform to make the product of all the elements in the array divisible by $2^n$. Note that such a set of operations does not always exist.

给你一个正整数数组 $a_1, a_2, \ldots, a_n$。

要求使数组中所有数的乘积（即 $a_1 \cdot a_2 \cdot \ldots \cdot a_n$）能被 $2^n$ 整除。

你可以执行以下操作任意多次：

• 任选一个下标 $i$（$1 \leq i \leq n$），并将 $a_i$ 替换为 $a_i = a_i \cdot i$。

但你不能对同一个下标重复执行该操作。换言之，所有被选中的 $i$ 值必须互不相同。

求使数组所有元素的乘积能被 $2^n$ 整除所需的最少操作次数。注意：这样的操作集合并不总是存在。

The first line of the input contains a single integer $t$ $(1 \leq t \leq 10^4$) — the number test cases.

Then the descriptions of the input data sets follow.

The first line of each test case contains a single integer $n$ ($1 \leq n \leq 2 \cdot 10^5$) — the length of array $a$.

The second line of each test case contains exactly $n$ integers: $a_1, a_2, \ldots, a_n$ ($1 \leq a_i \leq 10^9$).

It is guaranteed that the sum of $n$ values over all test cases in a test does not exceed $2 \cdot 10^5$.

输入的第一行包含一个整数 $t$（$1 \leq t \leq 10^4$）—— 表示测试用例的数量。

随后是各测试用例的输入数据描述。

每个测试用例的第一行包含一个整数 $n$（$1 \leq n \leq 2 \cdot 10^5$）—— 表示数组 $a$ 的长度。

每个测试用例的第二行包含恰好 $n$ 个整数：$a_1, a_2, \ldots, a_n$（$1 \leq a_i \leq 10^9$）。

保证所有测试用例中 $n$ 的总和不超过 $2 \cdot 10^5$。

For each test case, print the least number of operations to make the product of all numbers in the array divisible by $2^n$. If the answer does not exist, print -1.

对于每个测试用例，输出使数组中所有数的乘积能被 $2^n$ 整除所需的最少操作次数。如果不存在这样的答案，则输出 $-1$。

In the first test case, the product of all elements is initially $2$, so no operations needed.

In the second test case, the product of elements initially equals $6$. We can apply the operation for $i = 2$, and then $a_2$ becomes $2\cdot2=4$, and the product of numbers becomes $3\cdot4=12$, and this product of numbers is divided by $2^n=2^2=4$.

In the fourth test case, even if we apply all possible operations, we still cannot make the product of numbers divisible by $2^n$ — it will be $(13\cdot1)\cdot(17\cdot2)\cdot(1\cdot3)\cdot(1\cdot4)=5304$, which does not divide by $2^n=2^4=16$.

In the fifth test case, we can apply operations for $i = 2$ and $i = 4$.

在第一个测试用例中，所有元素的乘积初始为 $2$，因此无需进行任何操作。

在第二个测试用例中，元素的乘积初始为 $6$。我们可以对 $i = 2$ 执行操作，此时 $a_2$ 变为 $2\cdot2=4$，数字的乘积变为 $3\cdot4=12$，而该乘积可被 $2^n=2^2=4$ 整除。

在第四个测试用例中，即使执行所有可能的操作，我们仍无法使数字的乘积被 $2^n$ 整除——其结果为 $(13\cdot1)\cdot(17\cdot2)\cdot(1\cdot3)\cdot(1\cdot4)=5304$，而 $5304$ 不能被 $2^n=2^4=16$ 整除。

在第五个测试用例中，我们可以对 $i = 2$ 和 $i = 4$ 执行操作。