Superfactorial Numeral System

Interview

Time limit1sMemory limit1024 MB

Summary

Convert the fraction p/q into superfactorial digits a_1, a_2, ... where each a_k is below k and is weighted by 1/k!.

Level: Medium4 of 10
Topics: Math, Implementation
Solved: No attempts yet

Problem

On the most perfect planet i1c5l, programming contests use various numeral systems. In the second division, contestants use the superfactorial numeral system. In this system, a number is written as a linear combination of reciprocals of factorials.

$\frac{p}{q} = a_1 + \frac{a_2}{2!} + \frac{a_3}{3!} + \ldots + \frac{a_n}{n!}$

Here $a_1$ is a non-negative integer, and each integer $a_k$ with $k \ge 2$ satisfies $0 \le a_k < k$ . Nonsignificant zeros at the tail of a superfactorial representation are omitted. Find the representation of the rational number $\frac{p}{q}$ in the superfactorial numeral system.

Input

A single line contains two space-separated integers $p$ and $q$ ( $1 \le p \le 10^6$ , $1 \le q \le 10^6$ ).

Output

A single line with the space-separated integers $a_1, a_2, \ldots, a_n$ that form the superfactorial representation of $\frac{p}{q}$ . If several solutions exist, print any of them.

Examples3

Example 1
Input
```
1 2
```
Expected output
```
0 1
```
Example 2
Input
```
2 10
```
Expected output
```
0 0 1 0 4
```
Example 3
Input
```
10 2
```
Expected output
```
5
```