sum-of-k-mirror-numbers

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cangjie

Algorithms

A k-mirror number is a positive integer without leading zeros that reads the same both forward and backward in base-10 as well as in base-k.

<ul>
	<li>For example, <code>9</code> is a 2-mirror number. The representation of <code>9</code> in base-10 and base-2 are <code>9</code> and <code>1001</code> respectively, which read the same both forward and backward.</li>
	<li>On the contrary, <code>4</code> is not a 2-mirror number. The representation of <code>4</code> in base-2 is <code>100</code>, which does not read the same both forward and backward.</li>
</ul>

Given the base <code>k</code> and the number <code>n</code>, return the sum of the <code>n</code> smallest k-mirror numbers.

&nbsp;
Example 1:

<pre>
Input: k = 2, n = 5
Output: 25
Explanation:
The 5 smallest 2-mirror numbers and their representations in base-2 are listed as follows:
 base-10 base-2
 1 1
 3 11
 5 101
 7 111
 9 1001
Their sum = 1 + 3 + 5 + 7 + 9 = 25. 
</pre>

Example 2:

<pre>
Input: k = 3, n = 7
Output: 499
Explanation:
The 7 smallest 3-mirror numbers are and their representations in base-3 are listed as follows:
 base-10 base-3
 1 1
 2 2
 4 11
 8 22
 121 11111
 151 12121
 212 21212
Their sum = 1 + 2 + 4 + 8 + 121 + 151 + 212 = 499.
</pre>

Example 3:

<pre>
Input: k = 7, n = 17
Output: 20379000
Explanation: The 17 smallest 7-mirror numbers are:
1, 2, 3, 4, 5, 6, 8, 121, 171, 242, 292, 16561, 65656, 2137312, 4602064, 6597956, 6958596
</pre>

&nbsp;
Constraints:

<ul>
	<li><code>2 &lt;= k &lt;= 9</code></li>
	<li><code>1 &lt;= n &lt;= 30</code></li>
</ul>

kMirror

Sum of k-Mirror Numbers

Strobogrammatic Number II

strobogrammatic-number-ii

中心对称数 II

Prime Palindrome

prime-palindrome

回文质数

Math

Enumeration

一个 k 镜像数字&nbsp;指的是一个在十进制和 k 进制下从前往后读和从后往前读都一样的&nbsp;没有前导 0&nbsp;的&nbsp;正&nbsp;整数。

<ul>
	<li>比方说，<code>9</code>&nbsp;是一个 2 镜像数字。<code>9</code>&nbsp;在十进制下为&nbsp;<code>9</code>&nbsp;，二进制下为&nbsp;<code>1001</code>&nbsp;，两者从前往后读和从后往前读都一样。</li>
	<li>相反地，<code>4</code>&nbsp;不是一个 2 镜像数字。<code>4</code>&nbsp;在二进制下为&nbsp;<code>100</code>&nbsp;，从前往后和从后往前读不相同。</li>
</ul>

给你进制&nbsp;<code>k</code>&nbsp;和一个数字&nbsp;<code>n</code>&nbsp;，请你返回 k 镜像数字中 最小 的 <code>n</code>&nbsp;个数 之和&nbsp;。

&nbsp;

示例 1：

<pre>输入：k = 2, n = 5
输出：25
解释：
最小的 5 个 2 镜像数字和它们的二进制表示如下：
 十进制 二进制
 1 1
 3 11
 5 101
 7 111
 9 1001
它们的和为 1 + 3 + 5 + 7 + 9 = 25 。
</pre>

示例 2：

<pre>输入：k = 3, n = 7
输出：499
解释：
7 个最小的 3 镜像数字和它们的三进制表示如下：
 十进制 三进制
 1 1
 2 2
 4 11
 8 22
 121 11111
 151 12121
 212 21212
它们的和为 1 + 2 + 4 + 8 + 121 + 151 + 212 = 499 。
</pre>

示例 3：

<pre>输入：k = 7, n = 17
输出：20379000
解释：17 个最小的 7 镜像数字分别为：
1, 2, 3, 4, 5, 6, 8, 121, 171, 242, 292, 16561, 65656, 2137312, 4602064, 6597956, 6958596
</pre>

&nbsp;

提示：

<ul>
	<li><code>2 &lt;= k &lt;= 9</code></li>
	<li><code>1 &lt;= n &lt;= 30</code></li>
</ul>