2155. Addition Chains

单点时限: 2.0 sec

内存限制: 256 MB

An addition chain for is an integer sequence with the following four properties:

  • For each () there exist two (not necessarily different) integers and () with

You are given an integer . Your job is to construct an addition chain for with minimal length. If there is more than one such sequence, any one is acceptable.

For example, and are both valid solutions when you are asked for an addition chain for .

输入格式

The input will contain one or more test cases. Each test case consists of one line containing one integer (). Input is terminated by a value of zero (0) for .

输出格式

For each test case, print one line containing the required integer sequence. Separate the numbers by one blank.

样例

Input
5
7
12
15
77
0
Output
1 2 4 5
1 2 4 6 7
1 2 4 8 12
1 2 4 5 10 15
1 2 4 8 9 17 34 68 77

14 人解决,16 人已尝试。

20 份提交通过,共有 29 份提交。

3.9 EMB 奖励。

创建: 11 年,7 月前.

修改: 2 年前.

最后提交: 1 年,8 月前.

来源: Ulm Local 1997

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