Count the Number of Ideal Arrays
<p>You are given two integers <code>n</code> and <code>maxValue</code>, which are used to describe an <strong>ideal</strong> array.</p>

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

<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>

<p>Return <em>the number of <strong>distinct</strong> ideal arrays of length </em><code>n</code>. Since the answer may be very large, return it modulo <code>10<sup>9</sup> + 7</code>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> n = 2, maxValue = 5
<strong>Output:</strong> 10
<strong>Explanation:</strong> 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>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> n = 5, maxValue = 3
<strong>Output:</strong> 11
<strong>Explanation:</strong> 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 2<sup>nd</sup> 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 2<sup>nd</sup> 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>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>2 &lt;= n &lt;= 10<sup>4</sup></code></li>
	<li><code>1 &lt;= maxValue &lt;= 10<sup>4</sup></code></li>
</ul>
Math; Dynamic Programming; Combinatorics; Number Theory
func idealArrays(n int, maxValue int) (ans int) {
	const mod = int(1e9 + 7)
	c := make([][]int, n)
	for i := 0; i < n; i++ {
		c[i] = make([]int, 16)
		for j := 0; j <= i && j < 16; j++ {
			if j == 0 {
				c[i][j] = 1
			} else {
				c[i][j] = (c[i-1][j] + c[i-1][j-1]) % mod
			}
		}
	}

	f := make([][16]int, maxValue+1)
	for i := 1; i <= maxValue; i++ {
		f[i][1] = 1
	}
	for j := 1; j < 15; j++ {
		for i := 1; i <= maxValue; i++ {
			for k := 2; k*i <= maxValue; k++ {
				f[k*i][j+1] = (f[k*i][j+1] + f[i][j]) % mod
			}
		}
	}

	for i := 1; i <= maxValue; i++ {
		for j := 1; j < 16; j++ {
			ans = (ans + f[i][j]*c[n-1][j-1]) % mod
		}
	}
	return
}
