[seqfan] Augusto Santi's A351871
Robert Gerbicz
robert.gerbicz at gmail.com
Sun Sep 18 17:18:18 CEST 2022
a(1), a(2) are the first two (positive) integers then for n>2:
a(n)=g+(a(n-1)+(a-2))/g, where g=gcd(a(n-1),a(n-2)).
If a(1) and a(2) are odd then easily all numbers in the sequence are odd.
If a(1) or a(2) is even, then with induction for every two consecutive
numbers in the sequence at least one of them is even.
It gives that it is a partitioning of the sequences into 2 groups. And what
is interesting is that it looks like that in the first group the sequence
always goes to infinity, and in the second group it always goes to a cycle.
At least I have no counter-example for this conjecture.
And three more cycle lengths:
for a(1)=52, a(2)=378 the sequence starts with:
52, 378, 217, 92, 310, 203, 514, 718, 618, 670, 646, 660, 655, 268, 924,
302, 615, 918, 514, 718, ...
It gives a cycle length of 12, starting at 514.
for a(1)=264, a(2)=1037 the sequence starts with:
264, 1037, 1302, 2340, 613, 2954, 3568, 3263, 6832, 10096, 1074, 5587,
6662, 12250, 9458, 10856, 10159, 21016, 31176, 6532, 9431, 15964, 25396,
10344, 8939, 19284, 28224, 3971, 32196, 36168, 5709, 1302, 2340, ...
It gives a cycle length of 29, starting at 1302.
for a(1)=542, a(2)=6017 it gives a cycle of length 802, the maximum term
is 557981456058.
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