Sync LeetCode submission Runtime - 19 ms (96.83%), Memory - 25.1 MB (46.03%)

Author: jimit105

2415-count-the-number-of-ideal-arrays/README.md

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+<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>

2415-count-the-number-of-ideal-arrays/solution.py

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+MOD = 10**9 + 7
+MAX_N = 10**4 + 10
+MAX_P = 15  
+sieve = [0] * MAX_N  
+
+for i in range(2, MAX_N):
+    if sieve[i] == 0:
+        for j in range(i, MAX_N, i):
+            sieve[j] = i
+
+ps = [[] for _ in range(MAX_N)]
+
+for i in range(2, MAX_N):
+    x = i
+    while x > 1:
+        p = sieve[x]
+        cnt = 0
+        while x % p == 0:
+            x //= p
+            cnt += 1
+        ps[i].append(cnt)
+
+c = [[0] * (MAX_P + 1) for _ in range(MAX_N + MAX_P)]
+
+c[0][0] = 1
+for i in range(1, MAX_N + MAX_P):
+    c[i][0] = 1
+    for j in range(1, min(i, MAX_P) + 1):
+        c[i][j] = (c[i - 1][j] + c[i - 1][j - 1]) % MOD
+
+
+class Solution:
+    def idealArrays(self, n: int, maxValue: int) -> int:
+        ans = 0
+        for x in range(1, maxValue + 1):
+            mul = 1
+            for p in ps[x]:
+                mul = mul * c[n + p - 1][p] % MOD
+            ans = (ans + mul) % MOD
+        return ans