maximize-number-of-nice-divisors

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Algorithms

You are given a positive integer <code>primeFactors</code>. You are asked to construct a positive integer <code>n</code> that satisfies the following conditions:

<ul>
 <li>The number of prime factors of <code>n</code> (not necessarily distinct) is at most <code>primeFactors</code>.</li>
 <li>The number of nice divisors of <code>n</code> is maximized. Note that a divisor of <code>n</code> is nice if it is divisible by every prime factor of <code>n</code>. For example, if <code>n = 12</code>, then its prime factors are <code>[2,2,3]</code>, then <code>6</code> and <code>12</code> are nice divisors, while <code>3</code> and <code>4</code> are not.</li>
</ul>

Return the number of nice divisors of <code>n</code>. Since that number can be too large, return it modulo <code>109 + 7</code>.

Note that a prime number is a natural number greater than <code>1</code> that is not a product of two smaller natural numbers. The prime factors of a number <code>n</code> is a list of prime numbers such that their product equals <code>n</code>.

&nbsp;
Example 1:

<pre>
Input: primeFactors = 5
Output: 6
Explanation: 200 is a valid value of n.
It has 5 prime factors: [2,2,2,5,5], and it has 6 nice divisors: [10,20,40,50,100,200].
There is not other value of n that has at most 5 prime factors and more nice divisors.
</pre>

Example 2:

<pre>
Input: primeFactors = 8
Output: 18
</pre>

&nbsp;
Constraints:

<ul>
	<li><code>1 &lt;= primeFactors &lt;= 109</code></li>
</ul>

maxNiceDivisors

Maximize Number of Nice Divisors

Integer Break

integer-break

整数拆分

Recursion

Math

Number Theory

给你一个正整数 <code>primeFactors</code> 。你需要构造一个正整数 <code>n</code> ，它满足以下条件：

<ul>
	<li><code>n</code> 质因数（质因数需要考虑重复的情况）的数目 不超过 <code>primeFactors</code> 个。</li>
	<li><code>n</code> 好因子的数目最大化。如果 <code>n</code> 的一个因子可以被 <code>n</code> 的每一个质因数整除，我们称这个因子是 好因子 。比方说，如果 <code>n = 12</code> ，那么它的质因数为 <code>[2,2,3]</code> ，那么 <code>6</code> 和 <code>12</code> 是好因子，但 <code>3</code> 和 <code>4</code> 不是。</li>
</ul>

请你返回 <code>n</code> 的好因子的数目。由于答案可能会很大，请返回答案对 <code>109 + 7</code> 取余 的结果。

请注意，一个质数的定义是大于 <code>1</code> ，且不能被分解为两个小于该数的自然数相乘。一个数 <code>n</code> 的质因子是将 <code>n</code> 分解为若干个质因子，且它们的乘积为 <code>n</code> 。

 

示例 1：

<pre>
输入：primeFactors = 5
输出：6
解释：200 是一个可行的 n 。
它有 5 个质因子：[2,2,2,5,5] ，且有 6 个好因子：[10,20,40,50,100,200] 。
不存在别的 n 有至多 5 个质因子，且同时有更多的好因子。
</pre>

示例 2：

<pre>
输入：primeFactors = 8
输出：18
</pre>

 

提示：

<ul>
	<li><code>1 <= primeFactors <= 109</code></li>
</ul>