LeetCode #970 — MEDIUM

Powerful Integers

Move from brute-force thinking to an efficient approach using hash map strategy.

The Problem

Problem Statement

Given three integers x, y, and bound, return a list of all the powerful integers that have a value less than or equal to bound.

An integer is powerful if it can be represented as xⁱ + y^j for some integers i >= 0 and j >= 0.

You may return the answer in any order. In your answer, each value should occur at most once.

Example 1:

Input: x = 2, y = 3, bound = 10
Output: [2,3,4,5,7,9,10]
Explanation:
2 = 2⁰ + 3⁰
3 = 2¹ + 3⁰
4 = 2⁰ + 3¹
5 = 2¹ + 3¹
7 = 2² + 3¹
9 = 2³ + 3⁰
10 = 2⁰ + 3²

Example 2:

Input: x = 3, y = 5, bound = 15
Output: [2,4,6,8,10,14]

Constraints:

1 <= x, y <= 100
0 <= bound <= 10⁶

Step 01

Brute Force Baseline

Problem summary: Given three integers x, y, and bound, return a list of all the powerful integers that have a value less than or equal to bound. An integer is powerful if it can be represented as xi + yj for some integers i >= 0 and j >= 0. You may return the answer in any order. In your answer, each value should occur at most once.

Baseline thinking

Start with the most direct exhaustive search. That gives a correctness anchor before optimizing.

Pattern signal: Hash Map · Math

Example 1

2
3
10

Example 2

3
5
15

Core Insight

What unlocks the optimal approach

No official hints in dataset. Start from constraints and look for a monotonic or reusable state.

Interview move: turn each hint into an invariant you can check after every iteration/recursion step.

Step 03

Algorithm Walkthrough

Iteration Checklist

Define state (indices, window, stack, map, DP cell, or recursion frame).
Apply one transition step and update the invariant.
Record answer candidate when condition is met.
Continue until all input is consumed.

Use the first example testcase as your mental trace to verify each transition.

Step 04

Edge Cases

Minimum Input

Single element / shortest valid input

Validate boundary behavior before entering the main loop or recursion.

Duplicates & Repeats

Repeated values / repeated states

Decide whether duplicates should be merged, skipped, or counted explicitly.

Extreme Constraints

Upper-end input sizes

Re-check complexity target against constraints to avoid time-limit issues.

Invalid / Corner Shape

Empty collections, zeros, or disconnected structures

Handle special-case structure before the core algorithm path.

Step 05

Full Annotated Code

Source-backed implementations are provided below for direct study and interview prep.

// Accepted solution for LeetCode #970: Powerful Integers
class Solution {
    public List<Integer> powerfulIntegers(int x, int y, int bound) {
        Set<Integer> ans = new HashSet<>();
        for (int a = 1; a <= bound; a *= x) {
            for (int b = 1; a + b <= bound; b *= y) {
                ans.add(a + b);
                if (y == 1) {
                    break;
                }
            }
            if (x == 1) {
                break;
            }
        }
        return new ArrayList<>(ans);
    }
}

// Accepted solution for LeetCode #970: Powerful Integers
func powerfulIntegers(x int, y int, bound int) (ans []int) {
	s := map[int]struct{}{}
	for a := 1; a <= bound; a *= x {
		for b := 1; a+b <= bound; b *= y {
			s[a+b] = struct{}{}
			if y == 1 {
				break
			}
		}
		if x == 1 {
			break
		}
	}
	for x := range s {
		ans = append(ans, x)
	}
	return ans
}

# Accepted solution for LeetCode #970: Powerful Integers
class Solution:
    def powerfulIntegers(self, x: int, y: int, bound: int) -> List[int]:
        ans = set()
        a = 1
        while a <= bound:
            b = 1
            while a + b <= bound:
                ans.add(a + b)
                b *= y
                if y == 1:
                    break
            if x == 1:
                break
            a *= x
        return list(ans)

// Accepted solution for LeetCode #970: Powerful Integers
struct Solution;

impl Solution {
    fn powerful_integers(x: i32, y: i32, bound: i32) -> Vec<i32> {
        let mut set: Vec<bool> = vec![false; bound as usize + 1];
        let mut i = 0;
        while x.pow(i) < bound {
            let mut j = 0;
            while y.pow(j) < bound {
                let sum = x.pow(i) + y.pow(j);
                if sum <= bound {
                    set[sum as usize] = true;
                }
                j += 1;
                if y == 1 {
                    break;
                }
            }
            i += 1;
            if x == 1 {
                break;
            }
        }
        let mut res = vec![];
        for i in 0..=bound {
            if set[i as usize] {
                res.push(i);
            }
        }
        res
    }
}

#[test]
fn test() {
    let x = 2;
    let y = 3;
    let bound = 10;
    let res = vec![2, 3, 4, 5, 7, 9, 10];
    assert_eq!(Solution::powerful_integers(x, y, bound), res);
}

// Accepted solution for LeetCode #970: Powerful Integers
function powerfulIntegers(x: number, y: number, bound: number): number[] {
    const ans = new Set<number>();
    for (let a = 1; a <= bound; a *= x) {
        for (let b = 1; a + b <= bound; b *= y) {
            ans.add(a + b);
            if (y === 1) {
                break;
            }
        }
        if (x === 1) {
            break;
        }
    }
    return Array.from(ans);
}

Step 06

Interactive Study Demo

Use this to step through a reusable interview workflow for this problem.

Custom checkpoints (one per line)

Press Step or Run All to begin.

Step 07

Complexity Analysis

Time

O(log^2 bound)

Space

O(log^2 bound)

Approach Breakdown

BRUTE FORCE

O(n²) time

O(1) space

For each element, scan the rest of the array looking for a match. Two nested loops give n × (n−1)/2 comparisons = O(n²). No extra space since we only use loop indices.

HASH MAP

O(n) time

O(n) space

One pass through the input, performing O(1) hash map lookups and insertions at each step. The hash map may store up to n entries in the worst case. This is the classic space-for-time tradeoff: O(n) extra memory eliminates an inner loop.

Shortcut: Need to check “have I seen X before?” → hash map → O(n) time, O(n) space.

Coach Notes

Common Mistakes

Review these before coding to avoid predictable interview regressions.

Mutating counts without cleanup

Wrong move: Zero-count keys stay in map and break distinct/count constraints.

Usually fails on: Window/map size checks are consistently off by one.

Fix: Delete keys when count reaches zero.

Overflow in intermediate arithmetic

Wrong move: Temporary multiplications exceed integer bounds.

Usually fails on: Large inputs wrap around unexpectedly.

Fix: Use wider types, modular arithmetic, or rearranged operations.