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Ways to Express an Integer as Sum of Powers

Difficulty: Medium

Problem Description

Given two positive integers n and x, return the number of ways n can be expressed as the sum of the x-th power of unique positive integers. In other words, find the number of sets of unique integers [n1, n2, ..., nk] such that n = n1^x + n2^x + ... + nk^x. Since the result can be very large, return it modulo 10^9 + 7.

Key Insights

  • The problem involves representing a number as a sum of unique integers raised to a power.
  • Dynamic programming is a suitable approach to count the combinations effectively.
  • The solution requires iterating through potential bases and maintaining a state that keeps track of sums and counts.

Space and Time Complexity

Time Complexity: O(n * m), where n is the target integer and m is the maximum base integer that can contribute to the sum. Space Complexity: O(n), to store the number of ways to express sums up to n.

Solution

The problem can be solved using dynamic programming. We maintain an array dp where dp[i] represents the number of ways to express the integer i as the sum of unique integers raised to the power of x. The main idea is to iterate through all possible integers that can contribute to the sum, compute their x-th power, and update the dp array accordingly. We ensure that each integer is only used once by iterating backwards when updating the dp array.

Code Solutions

def countWays(n: int, x: int) -> int:
    MOD = 10**9 + 7
    dp = [0] * (n + 1)
    dp[0] = 1  # There's one way to sum to zero: use no numbers.
    
    # Iterate over possible bases
    for base in range(1, n + 1):
        power = base ** x
        if power > n:
            break
        # Update dp array backwards
        for j in range(n, power - 1, -1):
            dp[j] = (dp[j] + dp[j - power]) % MOD
            
    return dp[n]
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