(4^n-1)/3 is not square free for n = 9, 10, 18, 20, 21, 27, 30, 36, 40, 42, 45, 50, 54, 55, 60, 63, 68, 70, 72, 78, 80, 81, 84, .... On Wed, Jul 22, 2015 at 5:22 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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