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Interesting recurrence


Suppose that (a_n)_{n\geq 1} is a sequence of integers such that \displaystyle \sum_{d | n}a_d=2^n for all positive integers n. Prove that n | a_n for all n \geq 1.

Hint Try a combinatorial approach. Note that a_n is uniquely defined, and a_n can be taken as the number of  binary sequences of length d.

For more details see this question on Math SE

Or you can try it by induction. First note that a_1=2 and for p prime we have a_n=2^p-2. Then work with more factors inductively.

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