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Smallest representable

Source: Ukrainian Mathematical Olympiad 2024. Day 1, Problem 10.2

March 19, 2024
number theoryalgebrapower of 2

Problem Statement

You are given a positive integer nn. Find the smallest positive integer kk, for which there exist integers a1,a2,…,aka_1, a_2, \ldots, a_k, for which the following equality holds:
2a1+2a2+…+2ak=2n−n+k2^{a_1} + 2^{a_2} + \ldots + 2^{a_k} = 2^n - n + k
Proposed by Mykhailo Shtandenko
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