What is the optimal time complexity of solving Finding number of integers which has exactly x divisors?

The optimized solution achieves a time complexity of O(n) by using sorting, mathematical shortcuts, or hash table lookups.

What is the auxiliary space complexity for Finding number of integers which has exactly x divisors?

The space complexity is O(1) since we only use a minimal/constant amount of extra memory for tracking variables (or auxiliary space if dynamic structures are allocated).

What are the typical edge cases we must handle in Finding number of integers which has exactly x divisors?

Key edge cases include empty collections, negative or large bounds causing integer overflow, arrays with only one element, or inputs where target criteria are not met.

How can we optimize the brute force approach for Finding number of integers which has exactly x divisors?

Brute force methods often inspect all possible combinations. We optimize this by sorting the input list to enable binary search, or tracking processed items in a set to avoid redundant nested loop lookups.

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Finding number of integers which has exactly x divisors

Learn how to solve the 'Finding number of integers which has exactly x divisors' problem. This detailed resource details brute force and optimized approaches.

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Problem Statement

Easy

Write a function count_with_x_divisors(n, x) that takes two positive integers n and x, and returns the count of integers in the range [1, n] (inclusive) that have exactly x divisors.

A divisor of a number k is any integer that divides k evenly. For example, divisors of 6 are 1, 2, 3, 6 (4 divisors).

Constraints

•1 <= n <= 1000
•1 <= x <= 50

Examples

Example 1

Input

count_with_x_divisors(10, 2)

Output

Explanation

Numbers from 1 to 10 with exactly 2 divisors (i.e., prime numbers): 2, 3, 5, 7. That's 4 numbers.

Example 2

Input

count_with_x_divisors(10, 1)

Output

Explanation

Only the number 1 has exactly 1 divisor.

Example 3

Input

count_with_x_divisors(20, 4)

Output

Explanation

Numbers from 1-20 with exactly 4 divisors: 6(1,2,3,6), 8(1,2,4,8), 10(1,2,5,10), 14(1,2,7,14), 15(1,3,5,15). That's 5 numbers.

Need a Hint?

Use simple arithmetic operators (like modulo `%`, division `//`), conditional checks, or loops to inspect number properties.

Edge Cases to Watch

Empty list or null input variables
Single item lists/arrays
Extremely large input bounds causing integer or stack overflow

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