You are given 2 positive integers l and r. For any number x, all positive divisors of x except x are called the proper divisors of x.
A number is called special if it has exactly 2 proper divisors. For example:
Return the count of numbers in the range [l, r] that are not special.
Example 1:
Input: l = 5, r = 7
Output: 3
Explanation:
There are no special numbers in the range [5, 7].
Example 2:
Input: l = 4, r = 16
Output: 11
Explanation:
The special numbers in the range [4, 16] are 4 and 9.
Constraints:
1 <= l <= r <= 109To solve #3233 Find the Count of Numbers Which Are Not Special, observe the mathematical property of numbers considered special. A number is special if it has exactly two proper divisors. This only happens when the number is the square of a prime. For example, if n = p^2 where p is prime, its proper divisors are 1 and p.
Instead of checking each number in the range individually, reduce the problem to counting how many prime squares fall within the interval [l, r]. This means finding all primes p such that p^2 lies in the range. You can efficiently generate primes up to sqrt(r) using the Sieve of Eratosthenes, then count primes whose squares fall within the range.
Finally, subtract the count of special numbers from the total numbers in the range. This mathematical reduction avoids brute-force divisor checks and keeps the solution efficient.
| Approach | Time Complexity | Space Complexity |
|---|---|---|
| Sieve + Prime Square Counting | O(n log log n) where n = sqrt(r) | O(n) |
NeetCode
Use these hints if you're stuck. Try solving on your own first.
A special number must be a square of a prime number.
We need to find all primes in the range <code>[sqrt(l), sqrt(r)]</code>.
Use sieve to find primes till <code>sqrt(10<sup>9</sup>)</code>.
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Jot down your thoughts, approach, and key learnings
The Sieve of Eratosthenes is commonly used to generate all prime numbers up to sqrt(r). This allows us to quickly identify primes whose squares fall within the given range.
A number has exactly two proper divisors only when it is the square of a prime. For example, if n = p^2, its proper divisors are 1 and p. Any other number will either have fewer or more proper divisors.
The optimal approach is based on number theory. A number is special only if it is the square of a prime number. By generating primes up to sqrt(r) using the Sieve of Eratosthenes and counting prime squares in the range, we can efficiently compute the result.
Yes, problems like this are common in interviews because they test mathematical insight and optimization. Candidates must recognize patterns such as prime squares instead of relying on brute-force divisor counting.