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greatest odd divisor, limit of sequence

Source: Bulgaria 1985 P6

June 16, 2021
Sequencesalgebranumber theory

Problem Statement

Let αa\alpha_a denote the greatest odd divisor of a natural number aa, and let Sb=∑a=1bαaaS_b=\sum_{a=1}^b\frac{\alpha_a}a Prove that the sequence Sb/bS_b/b has a finite limit when b→∞b\to\infty, and find this limit.
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