So I was interested in a recurrence relation of which I wasn't able to find anything meaningful online about (that is, I couldn't solve it using Wolfram Alpha) Here's the recurrence relation: $a(n+1) = \sum_{i=1}^n a(i)/\gcd(i,n)$, where $a(1) = 1$. I was able to find the beginning sequence to be $1, 1, 3/2, 5/2, 29/8, 269/40, 1919/240$. There are a few question I want to answer: 1. Is there a way to find a general formula for this recurrence relation, and if not, then why? 2. Is there any way to measure or compare the growth rate of this relation to any other function? 3. Is there any previous work/research on relations like these that I can find because looking into this was very interesting.
2026-03-29 06:02:17.1774764137
Question related to finding a recurrence relation involving series and gcd
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Here is a plot of the first $100$ terms. Note that the $y$ axis is on a logarithmic scale. It appears that the sequence grows exponentially.