I just did a few Mma experiments on the sequence starting with f[1] = 0, f[2] = 1, and it doesn't appear to cycle. Well, at least through the first 1000 terms it was still getting net larger, though the sequence is clearly decreases at times. The 1000th term has 39 decimal digits: 705105353768951034109967840494543269698 Question: What is the probability that every (f[1], f[2]) in (Z+)^2 leads to a cycle: f[n] = f[n+p] for some p >= 1 and all sufficiently large n ? —Dan
On May 4, 2016, at 10:39 PM, Fred W. Helenius <fredh@ix.netcom.com> wrote:
On 5/4/2016 11:37 PM, James Propp wrote:
Must every sequence of squarefree positive integers satisfying the recurrence f(n) = SquarefreePart(f(n-1)+f(n-2)) eventually enter the cycle 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7, ...?
E.g., 7, 4, 11, 15, 26, 41, 67, 3, 70, 73, 143, 6, 149, 155, 19, 174, 193, 367, 35, 402, 437, 839, 319, 1158, 1477, 2635, 257, 723, 5, 182, 187, 41, 57, 2, 59, 61, 30, 91, 1, 23, 6, 29, 35, 1, 1, 2, 3, 5, 2, 7, ...
(This probably isn't new, but you never know.)
No, infinitely many other cycles are possible, including
n, 2n, 3n, 5n, 2n, 7n (period 6)
where n is any squarefree positive integer coprime to 210 = 2*3*5*7.
Less trivially, there are also the cycles
1 83 21 26 47 73 30 103 133 59 3 62 65 127 3 130 133 263 11 274 285 559 211 770 109 879 247 1126 1373 51 89 35 31 66 97 163 65 57 122 179 301 30 331 (period 43)
and
157 381 538 919 1457 66 1523 1589 778 263 1041 326 1367 1693 85 1778 23 1801 114 1915 2029 986 335 1321 46 1367 (period 26)
and
15 2162 2177 4339 181 1130 1311 2441 938 3379 4317 481 4798 5279 10077 3839 71 3910 3981 7891 742 8633 (period 22)
and also their multiples by suitable squarefree numbers.
-- Fred W. Helenius fredh@ix.netcom.com
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