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This approach involves iterating over possible powers of x and y, and calculating their sums while keeping the sum within the bound. We use two nested loops: one to iterate over powers of x and the other to iterate over powers of y. The results are stored in a set to ensure uniqueness.
Time Complexity: O(log(bound)^2) since the loops iterate logarithmically with respect to the bound.
Space Complexity: O(n) where n is the number of unique powerful integers within the bound.
1def powerful_integers(x, y, bound):
2 powerful = set()
3 i, j = 0, 0
4 while x**i <= bound:
5 j = 0
6 while x**i + y**j <= bound:
7 powerful.add(x**i + y**j)
8 if y == 1:
9 break
10 j += 1
11 if x == 1:
12 break
13 i += 1
14 return list(powerful)The function powerful_integers calculates all powerful integers within the given bound. It uses nested loops where the outer loop iterates over powers of x and the inner loop iterates over powers of y. It calculates the sum x^i + y^j and checks if it's within the bound. If so, it adds the sum to a set. This set ensures all elements are unique.
In this approach, we optimize the early termination condition by ensuring that we stop iterating as soon as additional powers do not contribute to any new sums within the bound. This prevents unnecessary iterations and computations.
Time Complexity: O(log(bound)^2) as it bounds progression over the log of the bound value.
Space Complexity: O(n) where n is the number of unique powerful integers.
1function powerfulIntegers(x, y, bound) {
2This JavaScript function employs a similar logic to previous approaches but uses multiplicative updates instead of powers for progression through x^i and y^j. It utilizes Set for uniqueness and converts the result to an array for the final output.