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To solve this problem, we need to find the minimal possible value of the sum of two numbers given the number of 1s in the binary representation of that sum.

Approach

The key insight here is understanding that the minimal number with exactly k 1s in its binary representation is a number consisting of k consecutive 1s starting from the least significant bit (rightmost bit). This number can be computed as 2^k -1 (since 2^k -1 in binary is 111...1 with k ones).

This is because any other number with k 1s in binary will have higher bits set, making it larger than 2^k -1. Additionally, any number S can always be written as the sum of two numbers (e.g., S = a + (S-a) for any a between 0 and S), so the sum constraint is always satisfied.

Solution Code

k = int(input())
print((1 << k) - 1)

Explanation

• For k (number of 1s in binary sum), the minimal sum is (1 << k) -1 where 1 << k is equivalent to 2^k (bitwise left shift).
• For example:
  • If k=1, the minimal sum is 1 (binary 1).
  • If k=2, the minimal sum is 3 (binary 11).
  • If k=3, the minimal sum is 7 (binary 111).

This approach efficiently computes the result in constant time O(1) using bitwise operations, which is optimal. The solution directly maps the input k to the desired minimal sum using the derived formula.

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