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largest odd number by which k is divisible, sum

Source: Polish MO Recond Round 1983 p4

September 9, 2024
number theorydivisorSum

Problem Statement

Let a(k) a(k) be the largest odd number by which k k is divisible. Prove that ∑k=12na(k)=13(4n+2). \sum_{k=1}^{2^n} a(k) = \frac{1}{3}(4^n+2).
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