Factorials and Powers of Two

3000ms

262144K

Description:

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 $$$k$$$.

Input:

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 $$$n$$$ ($$$1\le n\le 10^{12}$$$).

Output:

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 $$$k$$$.

Sample Input:

4
7
11
240
17179869184

Sample Output:

Note:

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 $$$k=1$$$.

Informação

Provedor Codeforces

Código CF1646C