Precisely one of your two conjectures is correct. For every odd positive integer k, we have 4^phi(k) - 1 is divisible by k (by Euler-Fermat). Hence 4^phi(3k) - 1 is divisible by 3k (since 3k is an odd positive integer!), so s_phi(3k) is divisible by k. In particular, s_18 = 22906492245 is divisible by 9 and thus isn't squarefree. As for each s_n having a new prime factor, that is a special case of a more general result called Zsigmondy's theorem: https://en.wikipedia.org/wiki/Zsigmondy%27s_theorem Sincerely, Adam P. Goucher
Sent: Wednesday, July 22, 2015 at 11:22 PM From: "Dan Asimov" <asimov@msri.org> To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] Number theory pattern?
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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