A14936. Factorials and Powers of Two

A number is called powerful if it is a power of two or a factorial. In other words, the number $m$ is powerful if there exists a non-negative integer $d$ such that $m=2^d$ or $m=d!$ , where $d!=1\cdot 2\cdot \ldots \cdot d$ (in particular, $0! = 1$ ). For example $1$ , $4$ , and $6$ are powerful numbers, because $1=1!$ , $4=2^2$ , and $6=3!$ but $7$ , $10$ , or $18$ are not.

You are given a positive integer $n$ . Find the minimum number $k$ such that $n$ can be represented as the sum of $k$ distinct powerful numbers, or say that there is no such $k$ .

Each test contains multiple test cases. The first line contains the number of test cases $t$ ( $1 \le t \le 100$ ). Description of the test cases follows.

A test case consists of only one line, containing one integer $n$ ( $1\le n\le 10^{12}$ ).

For each test case print the answer on a separate line.

If $n$ can not be represented as the sum of distinct powerful numbers, print $-1$ .

Otherwise, print a single positive integer — the minimum possible value of $k$ .

In the first test case, $7$ can be represented as $7=1+6$ , where $1$ and $6$ are powerful numbers. Because $7$ is not a powerful number, we know that the minimum possible value of $k$ in this case is $k=2$ .

In the second test case, a possible way to represent $11$ as the sum of three powerful numbers is $11=1+4+6$ . We can show that there is no way to represent $11$ as the sum of two or less powerful numbers.

In the third test case, $240$ can be represented as $240=24+32+64+120$ . Observe that $240=120+120$ is not a valid representation, because the powerful numbers have to be distinct.

In the fourth test case, $17179869184=2^{34}$ , so $17179869184$ is a powerful number and the minimum $k$ in this case is $k=1$ .