Prime Number of Set Bits in Binary Representation

LeetCode 767 | Difficulty: Easy

Easy

Problem Description

Given two integers left and right, return the count of numbers in the inclusive range [left, right] having a prime number of set bits in their binary representation.

Recall that the number of set bits an integer has is the number of 1's present when written in binary.

- For example, `21` written in binary is `10101`, which has `3` set bits.

Example 1:

Input: left = 6, right = 10
Output: 4
Explanation:
6  -> 110 (2 set bits, 2 is prime)
7  -> 111 (3 set bits, 3 is prime)
8  -> 1000 (1 set bit, 1 is not prime)
9  -> 1001 (2 set bits, 2 is prime)
10 -> 1010 (2 set bits, 2 is prime)
4 numbers have a prime number of set bits.

Example 2:

Input: left = 10, right = 15
Output: 5
Explanation:
10 -> 1010 (2 set bits, 2 is prime)
11 -> 1011 (3 set bits, 3 is prime)
12 -> 1100 (2 set bits, 2 is prime)
13 -> 1101 (3 set bits, 3 is prime)
14 -> 1110 (3 set bits, 3 is prime)
15 -> 1111 (4 set bits, 4 is not prime)
5 numbers have a prime number of set bits.

Constraints:

- `1 <= left <= right <= 10^6`

- `0 <= right - left <= 10^4`

Topics: Math, Bit Manipulation

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Approach

Bit Manipulation

Operate directly on binary representations. Key operations: AND (&), OR (|), XOR (^), NOT (~), shifts (<<, >>). XOR is especially useful: a ^ a = 0, a ^ 0 = a.

When to use

Finding unique elements, power of 2 checks, subset generation, toggling flags.

Mathematical

Look for mathematical patterns or formulas. Consider: modular arithmetic, GCD/LCM, prime factorization, combinatorics, or geometric properties.

When to use

Problems with clear mathematical structure, counting, number properties.

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Solutions

Solution 1: C# (Best: 82 ms)

Metric	Value
Runtime	82 ms
Memory	25.8 MB
Date	2022-01-19

Solution

public class Solution {
    public int CountPrimeSetBits(int left, int right) {
        List<int> primes = new List<int>() {2, 3, 5, 7, 11, 13, 17, 19};
    int count = 0;
    for (int i = left; i <= right; i++)
    {
        int bits = CountSetBits(i);
        if(primes.Contains(bits)) count++;      
    }
    return count;

}

private int CountSetBits(int n)
{
    int count = 0;
    while(n > 0)
    {
        if((n&1) == 1) count++;
        n = n >> 1;     
    }
    return count;
}
}

📜 1 more C# submission(s)

Submission (2022-01-19) — 97 ms, 25.5 MB

public class Solution {
    public int CountPrimeSetBits(int left, int right) {
        bool[] primes = new bool[33];
    primes[0] = false; primes[1] = false;
    primes[2] = true; primes[3] = true;
    for (int i = 4; i <= 32; i++)
    {
        primes[i] = true;
    }
    for (int i = 2; i*i <= 32; i++)
    {
        if(primes[i])
        {
            for (int j = i*2; j <= 32; j=j+i)
            {
                primes[j] = false;
            }
        }
    }
    int count = 0;
    for (int i = left; i <= right; i++)
    {
        int bits = CountSetBits(i);
        if(primes[bits]) count++;       
    }
    return count;

}

private int CountSetBits(int n)
{
    int count = 0;
    while(n > 0)
    {
        if((n&1) == 1) count++;
        n = n >> 1;     
    }
    return count;
}

}

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Complexity Analysis

Approach	Time	Space
Bit Manipulation	$O(n) or O(1)$	$O(1)$

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Interview Tips

Key Points

• Start by clarifying edge cases: empty input, single element, all duplicates.
• LeetCode provides 1 hint(s) for this problem — try solving without them first.

💡 Hints

Hint 1: Write a helper function to count the number of set bits in a number, then check whether the number of set bits is 2, 3, 5, 7, 11, 13, 17 or 19.