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 4 is special because its proper divisors are 1 and 2 (exactly 2 divisors)
  • The number 6 is not special because its proper divisors are 1, 2, 3 (3 divisors)
  • The number 9 is special because its proper divisors are 1 and 3 (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
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
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
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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Special Numbers VisualizationPattern Recognition:492549...💡 Key InsightA number has exactly 2 proper divisors ⟺ It's the square of a primeIf n = p² (p prime), then divisors of n are: 1, p, p² → proper divisors: 1, pAlgorithm:1. Find PrimesSieve up to √rO(√r log log √r)2. Check Squaresp² in [l, r]?O(π(√r))3. Count Non-SpecialTotal - Special(r-l+1) - count
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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