s_9 = 87381 = 3^2 7^1 19^1 73^1. In general, I think, if p | s_n, then p | s_kn for all k, and in particular, p^2 | s_np. As for the novel factor, I think that's very likely true. I suspect, in fact, that all non-novel factors are instances of the lemmas I just gave, and they numerically can never get up into the 4^n range, so novel factors must take up the slack. On Wed, Jul 22, 2015 at 6:22 PM, Dan Asimov <asimov@msri.org> wrote:
Consider the sequence s_n := (4^n-1)/3, n = 1,2,3,....
Back of the envelope shows that at least for very low n, s_n is squarefree and always has a prime factor that's not a factor of any previous s_n.
Do these patterns continue forever, and if so, why?
This is OEIS A002450 <https://oeis.org/A002450>, but these features are not mentioned there — so it seems likely they're both false.
—Dan
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