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A360209 Lexicographically earliest infinite sequence of distinct positive numbers such that, for n > 2, a(n) shares a factor with a(n-2) + a(n-1) but shares no factor with a(n-2).

%I A360209 #20 Mar 17 2023 07:35:55
%S A360209 1,2,3,5,4,6,15,7,8,9,17,10,12,11,23,14,37,27,16,43,59,18,21,13,20,22,
%T A360209 33,25,26,24,35,295,32,36,51,29,28,19,47,30,44,259,39,34,73,107,38,40,
%U A360209 45,119,41,46,42,55,97,48,50,49,57,52,109,63,54,65,77,56,76,69,75,58,91,149,60,66
%N A360209 Lexicographically earliest infinite sequence of distinct positive numbers such that, for n > 2, a(n) shares a factor with a(n-2) + a(n-1) but shares no factor with a(n-2).
%C A360209 To ensure the sequence is infinite another criterion must be satisfied when choosing a(n), namely a(n) + a(n-1) must contain a factor not in a(n-1). If this were not the case, a(n+1) = a(n) + a(n-1) would share a factor with both a(n) + a(n-1) and a(n-1), terminating the sequence.
%C A360209 In the first 100000 terms the fixed points for n > 2 are 3, 6, 441, 1677, 3629, 9701, 17131, although it is likely more exist. The sequence is conjectured to be a permutation of the positive integers.
%H A360209 Scott R. Shannon, <a href="/A360209/a360209_1.png">Image for n=1..100000</a>. The green line is a(n) = n.
%e A360209 a(7) = 15 as a(5) + a(6) = 4 + 6 = 10, and 15 is the smallest positive unused number that shares a factor with 10 but not with a(5) = 4.
%e A360209 a(41) = 44 as a(39) + a(40) = 47 + 30 = 77, and 44 shares a factor with 77 but not with a(39) = 47. Note that 42 also satisfies these criteria but 30 + 42 = 72 which shares all its factors with a(40) = 30, thus setting a(41) = 42 would make it impossible to find a(42).
%Y A360209 Cf. A251604 (does not share with a(n-1)), A098550, A336957, A337136, A359557, A353239.
%K A360209 nonn
%O A360209 1,2
%A A360209 _Scott R. Shannon_, Jan 29 2023
%E A360209 a(6) and above corrected by _Scott R. Shannon_, Mar 17 2023