Find the Count of Numbers Which Are Not Special - Problem
You are given two positive integers l and r representing a range. Your task is to find how many numbers in the range [l, r] are not special.
What makes a number special?
A number is called special if it has exactly 2 proper divisors. The proper divisors of a number x are all positive divisors of x except x itself.
Examples:
- The number
4is special because its proper divisors are1and2(exactly 2 divisors) - The number
6is not special because its proper divisors are1, 2, 3(3 divisors) - The number
9is special because its proper divisors are1and3(exactly 2 divisors)
Key Insight: A number has exactly 2 proper divisors if and only if it's the square of a prime number! This is because if n = p² where p is prime, then the divisors of n are 1, p, p², making the proper divisors 1 and p.
Input & Output
example_1.py — Basic Range
$
Input:
l = 5, r = 7
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Output:
3
💡 Note:
Numbers in range [5,7] are: 5, 6, 7. None of these are special (5 has proper divisors [1], 6 has proper divisors [1,2,3], 7 has proper divisors [1]). So all 3 numbers are not special.
example_2.py — Range with Special Numbers
$
Input:
l = 4, r = 16
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Output:
11
💡 Note:
Numbers in range [4,16]: 4,5,6,7,8,9,10,11,12,13,14,15,16 (13 total). Special numbers: 4 (2²), 9 (3²) (2 special). Non-special: 13 - 2 = 11.
example_3.py — Single Number
$
Input:
l = 4, r = 4
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Output:
0
💡 Note:
Only number 4 in range. 4 = 2² is special (proper divisors: 1, 2). So 0 numbers are not special.
Constraints
- 1 ≤ l ≤ r ≤ 109
- The range [l, r] contains at most 105 numbers
- Special numbers are squares of prime numbers
Visualization
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Understanding the Visualization
1
Identify Pattern
Special numbers: 4, 9, 25, 49... all are squares of primes!
2
Generate Primes
Use Sieve of Eratosthenes to find all primes up to √r
3
Count Prime Squares
Check which prime squares fall in range [l, r]
Key Takeaway
🎯 Key Insight: Special numbers are exactly squares of prime numbers, allowing us to solve this efficiently using prime generation instead of checking divisors for each number.
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Explanation
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