A14678. Paprika and Permutation

Paprika loves permutations. She has an array $a_1, a_2, \dots, a_n$ . She wants to make the array a permutation of integers $1$ to $n$ .

In order to achieve this goal, she can perform operations on the array. In each operation she can choose two integers $i$ ( $1 \le i \le n$ ) and $x$ ( $x > 0$ ), then perform $a_i := a_i \bmod x$ (that is, replace $a_i$ by the remainder of $a_i$ divided by $x$ ). In different operations, the chosen $i$ and $x$ can be different.

Determine the minimum number of operations needed to make the array a permutation of integers $1$ to $n$ . If it is impossible, output $-1$ .

A permutation is an array consisting of $n$ distinct integers from $1$ to $n$ in arbitrary order. For example, $[2,3,1,5,4]$ is a permutation, but $[1,2,2]$ is not a permutation ( $2$ appears twice in the array) and $[1,3,4]$ is also not a permutation ( $n=3$ but there is $4$ in the array).

Each test contains multiple test cases. The first line contains a single integer $t$ ( $1 \le t \le 10^4$ ) — the number of test cases. Description of the test cases follows.

The first line of each test case contains an integer $n$ ( $1 \le n \le 10^5$ ).

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

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

For each test case, output the minimum number of operations needed to make the array a permutation of integers $1$ to $n$ , or $-1$ if it is impossible.

For the first test, the only possible sequence of operations which minimizes the number of operations is:

- Choose $i=2$ , $x=5$ . Perform $a_2 := a_2 \bmod 5 = 2$ .

For the second test, it is impossible to obtain a permutation of integers from $1$ to $n$ .