count-the-number-of-ideal-arrays

Count the Number of Ideal Arrays

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idealArrays

You are given two integers <code>n</code> and <code>maxValue</code>, which are used to describe an ideal array.

A 0-indexed integer array <code>arr</code> of length <code>n</code> is considered ideal if the following conditions hold:

<ul>
	<li>Every <code>arr[i]</code> is a value from <code>1</code> to <code>maxValue</code>, for <code>0 &lt;= i &lt; n</code>.</li>
	<li>Every <code>arr[i]</code> is divisible by <code>arr[i - 1]</code>, for <code>0 &lt; i &lt; n</code>.</li>
</ul>

Return the number of distinct ideal arrays of length <code>n</code>. Since the answer may be very large, return it modulo <code>109 + 7</code>.

&nbsp;
Example 1:

<pre>
Input: n = 2, maxValue = 5
Output: 10
Explanation: The following are the possible ideal arrays:
- Arrays starting with the value 1 (5 arrays): [1,1], [1,2], [1,3], [1,4], [1,5]
- Arrays starting with the value 2 (2 arrays): [2,2], [2,4]
- Arrays starting with the value 3 (1 array): [3,3]
- Arrays starting with the value 4 (1 array): [4,4]
- Arrays starting with the value 5 (1 array): [5,5]
There are a total of 5 + 2 + 1 + 1 + 1 = 10 distinct ideal arrays.
</pre>

Example 2:

<pre>
Input: n = 5, maxValue = 3
Output: 11
Explanation: The following are the possible ideal arrays:
- Arrays starting with the value 1 (9 arrays): 
 - With no other distinct values (1 array): [1,1,1,1,1] 
 - With 2nd distinct value 2 (4 arrays): [1,1,1,1,2], [1,1,1,2,2], [1,1,2,2,2], [1,2,2,2,2]
 - With 2nd distinct value 3 (4 arrays): [1,1,1,1,3], [1,1,1,3,3], [1,1,3,3,3], [1,3,3,3,3]
- Arrays starting with the value 2 (1 array): [2,2,2,2,2]
- Arrays starting with the value 3 (1 array): [3,3,3,3,3]
There are a total of 9 + 1 + 1 = 11 distinct ideal arrays.
</pre>

&nbsp;
Constraints:

<ul>
	<li><code>2 &lt;= n &lt;= 104</code></li>
	<li><code>1 &lt;= maxValue &lt;= 104</code></li>
</ul>

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