On 5/7/2016 12:28 AM, Dan Asimov wrote:
(*) f(N) = sk( f(N-1) + f(N-2) ), N >= 2
[where sk(n) is the squarefree kernel, or radical, of n]
Question: --------- Are there any positive integers f(0) and f(1) that lead to a periodic sequence, as defined by (*) above?
I already mentioned some simple examples that arise where f(0) and f(1) are not coprime: 2, 2 (period 1) 3, 3, 6, 3, 3, 6 (period 3) 5, 10, 15, 5, 10, 15 (period 3) There are many others; in particular, f(0) = f(1) = 2k (k odd, squarefree) always generates a constant sequence and any two initial values that are both even will result in an eventually constant sequence (an easy exercise). On the other hand, it seems that sequences with a common factor which is a larger prime may grow indefinitely. f(0) = f(1) = 17 seems like a borderline example; it has gradual rises mixed with teasing falls, but (for 2000 terms, at least) achieves net growth (up to 52 digits). I have since found three examples of periodic sequences starting with coprime initial values. Here they are: 15 146 161 307 78 385 463 106 569 (period 9) 222 1589 1811 170 1981 717 2698 3415 6113 2382 8495 10877 9686 20563 10083 30646 3133 33779 4614 38393 43007 4070 47077 17049 64126 16235 26787 6146 32933 39079 36006 75085 111091 5818 5083 10901 (period 46) 770 559 1329 118 1447 1565 1506 3071 4577 478 5055 5533 5294 1203 6497 (period 15) -- Fred W. Helenius fredh@ix.netcom.com