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

Description

You are given two integers n and k.

For any positive integer x, define the following sequence:

  • p0 = x
  • pi+1 = popcount(pi) 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 pd = 1.

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:

x Binary Sequence
2 "10" 2 → 1
4 "100" 4 → 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:

x Binary Sequence
3 "11" 3 → 2 → 1
5 "101" 5 → 2 → 1
6 "110" 6 → 2 → 1

Thus, the answer is 3.

 

Constraints:

  • 1 <= n <= 1015
  • 0 <= k <= 5

Solutions

Solution 1

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1


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