A11944 | Summarize to the Power of Two

A sequence $a_1, a_2, \dots, a_n$ is called good if, for each element $a_i$ , there exists an element $a_j$ ( $i \ne j$ ) such that $a_i+a_j$ is a power of two (that is, $2^d$ for some non-negative integer $d$ ).

For example, the following sequences are good:

- $[5, 3, 11]$ (for example, for $a_1=5$ we can choose $a_2=3$ . Note that their sum is a power of two. Similarly, such an element can be found for $a_2$ and $a_3$ ),
- $[1, 1, 1, 1023]$ ,
- $[7, 39, 89, 25, 89]$ ,
- $[]$ .

Note that, by definition, an empty sequence (with a length of $0$ ) is good.

For example, the following sequences are not good:

- $[16]$ (for $a_1=16$ , it is impossible to find another element $a_j$ such that their sum is a power of two),
- $[4, 16]$ (for $a_1=4$ , it is impossible to find another element $a_j$ such that their sum is a power of two),
- $[1, 3, 2, 8, 8, 8]$ (for $a_3=2$ , it is impossible to find another element $a_j$ such that their sum is a power of two).

You are given a sequence $a_1, a_2, \dots, a_n$ . What is the minimum number of elements you need to remove to make it good? You can delete an arbitrary set of elements.

The first line contains the integer $n$ ( $1 \le n \le 120000$ ) — the length of the given sequence.

The second line contains the sequence of integers $a_1, a_2, \dots, a_n$ ( $1 \le a_i \le 10^9$ ).

Print the minimum number of elements needed to be removed from the given sequence in order to make it good. It is possible that you need to delete all $n$ elements, make it empty, and thus get a good sequence.