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Define radical of an integer Wiki

$$\displaystyle{\mathrm{rad}}(n)=\prod_{{\scriptstyle p\mid n\atop p\:{\text{prime}}}}p$$

Example $n=504=2^3\cdot3^2\cdot7$ therefore ${\displaystyle \operatorname{rad}(504)=2\cdot3\cdot7=42}$

Define recurrence relation $a_k=\operatorname{rad}(a_{k-1}+a_{k-2})$ for $k\ge 2$ where $a_0=0,a_1=1$.

Sequence as https://oeis.org/A165911

$\{a_0,a_1,a_2,...\} = \{0, 1, 1, 2, 3, 5, 2, 7, 3, 10, 13, 23, 6, 29, 35, 2, 37, 39, 38,...\}$

It is not known that sequence stuck in any loop or period.

Is it possible to solve this problem? Also how to get to that solution? And would like to know a sequence that helps as a reference.


Comment from oeis

Through n=1688, this sequence does not loop. Does it grow indefinitely, or is it eventually periodic?

The graph suggests that the sequence had a chance to go into a cycle between terms 100 and 150, but by the time we get to 1688 terms the sequence seems to have reached escape velocity and there is no further hope of this happening. (Of course this is not a rigorous argument.) - N. J. A. Sloane, May 06 2016.


A few days ago, I rediscovered this sequence and didn’t do any progressive work on it but it’s interesting to me. Apologies for the immaturity and thank you in advance.

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