Prime Number of Set Bits in Binary Representation

Prime Number of Set Bits in Binary Representation - Problem

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.

Input & Output

Example 1 — Basic Range

$ Input: left = 6, right = 10

› Output: 4

💡 Note: 6 = 110₂ (2 bits, prime), 7 = 111₂ (3 bits, prime), 8 = 1000₂ (1 bit, not prime), 9 = 1001₂ (2 bits, prime), 10 = 1010₂ (2 bits, prime). So 4 numbers have prime set bits.

Example 2 — Small Range

$ Input: left = 10, right = 15

› Output: 5

💡 Note: 10 = 1010₂ (2 bits), 11 = 1011₂ (3 bits), 12 = 1100₂ (2 bits), 13 = 1101₂ (3 bits), 14 = 1110₂ (3 bits), 15 = 1111₂ (4 bits, not prime). Numbers 10,11,12,13,14 have prime set bits.

Example 3 — Single Number

$ Input: left = 4, right = 4

› Output: 0

💡 Note: 4 = 100₂ has 1 set bit, and 1 is not prime, so result is 0.

Constraints

1 ≤ left ≤ right ≤ 10⁶
0 ≤ right - left ≤ 10⁴

Visualization

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Understanding the Visualization

Input Range

Given left=6, right=10

Convert & Count

Convert each number to binary and count 1-bits

Check Prime

Count numbers where set bit count is prime

Key Takeaway

🎯 Key Insight: Only need to check if bit counts 1-32 are prime since integers have at most 32 bits

Asked in

G Google 15 f Facebook 12

The key insight is that integers have at most 32 bits, so we only need to check primes up to 32. Best approach is precomputed prime lookup for O(1) prime checking. Time: O(n × log m), Space: O(1)

Common Approaches

Approach	Time	Space	Notes
✓ Brute Force	O(n × log m × √k)	O(1)	Check each number, count set bits, verify if count is prime
Precomputed Primes	O(n × log m)	O(1)	Use precomputed prime set for O(1) prime checking

Brute Force — Algorithm Steps

Iterate through each number from left to right
Count set bits using bit manipulation
Check if the count is prime
Increment result if prime

Visualization

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Step-by-Step Walkthrough

Convert to Binary

Convert each number to binary representation

Count Set Bits

Count the number of 1s in binary

Check Prime

Verify if the count is a prime number

Code -

solution.c — C

#include <stdio.h>
#include <stdbool.h>
#include <math.h>

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

bool isPrime(int n) {
    if (n < 2) return false;
    if (n == 2) return true;
    if (n % 2 == 0) return false;
    for (int i = 3; i * i <= n; i += 2) {
        if (n % i == 0) return false;
    }
    return true;
}

int solution(int left, int right) {
    int result = 0;
    for (int num = left; num <= right; num++) {
        int setBitCount = countSetBits(num);
        if (isPrime(setBitCount)) {
            result++;
        }
    }
    return result;
}

int main() {
    int left, right;
    scanf("%d", &left);
    scanf("%d", &right);
    printf("%d\n", solution(left, right));
    return 0;
}

Time & Space Complexity

Time Complexity

⏱️

O(n × log m × √k)

n numbers in range, log m time to count bits, √k time for primality test where k is max bit count

⚡ Linearithmic

Space Complexity

O(1)

Only using constant extra variables

✓ Linear Space

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Medium Frequency

~15 min Avg. Time

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Smart Actions

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Input & Output

Constraints

Visualization

Related Problems

Common Approaches

Brute Force — Algorithm Steps

Visualization

Code -

Time & Space Complexity

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