Let sk(N) mean the squarefree kernel (aka radical) of the integer N > 0. I.e., sk(N) = the product of all distinct prime factors of N. Then as regards the sequence defined by ----------------------------------------------- f(0) = 0, f(1) = 1, (*) f(N) = sk( f(N-1) + f(N-2) ), N >= 2 ----------------------------------------------- , precisely because as of 2009, this very sequence was already in OEIS, from Franklin T. Adams-Watters — I don't think it's appropriate that anyone else's name be associated with it. (Unless of course someone else discussed this even earlier.) I don't want to be immortalized in mathematics along with Kirmse <https://books.google.com/books?id=p4o-Uf-i-IUC&pg=PA26&lpg=PA26&dq=kirmse's+mistake&source=bl&ots=WB1X-E17tp&sig=2HBv_kW6BaI_A08soW1vizzaWOc&hl=en&sa=X&ved=0ahUKEwj20d_hj8fMAhVL9GMKHbmdCaoQ6AEINjAD#v=onepage&q=kirmse's%20mistake&f=false> by something called "Asimov's mistake" !!! * * * On the other hand, I guess the sequence defined by (*) but with arbitrary starting values f(0), f(1) is up for grabs. Question: --------- Are there any positive integers f(0) and f(1) that lead to a periodic sequence, as defined by (*) above? ((( And of course this generalizes to the same idea but with (*) replaced by (**) f(n) = sk( f(N-1) + ... + f(N-K) ), N >= K , with starting values supplied for f(1), ..., f(K). ))) —Dan
On May 6, 2016, at 2:31 PM, Neil Sloane <njasloane@gmail.com> wrote:
The classical "Pisano periods" (A001175) (an easily remembered name from the 1960's) give the period of Fib(n) mod n, and one can see that Fib_n has period 3 mod 2, period 8 mod 3, period 6 mod 4, but after that the periods get larger. So probably only the mod 2 effect will be significant