3621 - Number of Integers With Popcount-Depth Equal to K I.

Hard

You are given two integers n and k.

For any positive integer x, define the following sequence:

• <code>p₀ = x</code>

• <code>p_i+1 = popcount(p_i)</code> for all i >= 0, where popcount(y) is the number of set bits (1's) in the binary representation of y.

This sequence will eventually reach the value 1.

The popcount-depth of x is defined as the smallest integer d >= 0 such that <code>p_d = 1</code>.

For example, if x = 7 (binary representation "111"). Then, the sequence is: 7 → 3 → 2 → 1, so the popcount-depth of 7 is 3.

Your task is to determine the number of integers in the range [1, n] whose popcount-depth is exactly equal to k.

Return the number of such integers.

Example 1:

Input: n = 4, k = 1

Output: 2

Explanation:

The following integers in the range [1, 4] have popcount-depth exactly equal to 1:

Thus, the answer is 2.

Example 2:

Input: n = 7, k = 2

Output: 3

Explanation:

The following integers in the range [1, 7] have popcount-depth exactly equal to 2:

Thus, the answer is 3.

Constraints:

• <code>1 <= n <= 10¹⁵</code>

• 0 <= k <= 5