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%I #19 Apr 14 2022 10:37:33
%S 1,2,4,6,3,8,10,5,12,9,14,7,16,18,15,20,22,11,24,21,26,13,28,30,25,27,
%T 33,36,32,34,17,38,19,40,35,42,39,44,46,23,48,45,50,52,54,51,56,49,58,
%U 29,60,55,57,63,66,62,64,68,70,65,72,69,74,37,76,78,75,80,82,41,84,77,81,87,90,85,86
%N a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that has not appeared that shares a factor with max(a(n-2),a(n-1)).
%C Although all primes likely appear they do not occur in their natural order, e.g., 37 appears before 31. In the range studied each time a prime appears, beyond the initial 2, the previous term is a multiple of the same prime. The largest multiple in the first 500000 terms is six, first occurring at a(7782) = 8286, a(7783) = 1381. It is unknown if this ratio is unbounded for large n. As a prime term is less than its previous term the term following the prime will share a factor with previous multiple of the prime. This factor appears to always be a factor of the multiple and thus the term is not another multiple of the prime.
%C In the first 500000 terms the fixed points are 1, 2, 15, 25, 35. It is likely no more exist. The sequence is conjectured to be a permutation of the positive integers.
%H Michael De Vlieger, <a href="/A352976/b352976.txt">Table of n, a(n) for n = 1..10000</a>
%H Michael De Vlieger, <a href="/A352976/a352976_1.png">Annotated log-log scatterplot of a(n)</a>, n = 1..2^14, showing records in red and local minima in blue, highlighting primes in green and fixed points in amber.
%H Scott R. Shannon, <a href="/A352976/a352976.png">Image of the first 100000 terms</a>. The green line is y = n.
%e a(4) = 6 as max(a(2),a(3)) = max(2,4) = 4, and 6 is the smallest unused number that shares a factor with 4.
%e a(5) = 3 as max(a(3),a(4)) = max(4,6) = 6, and 3 is the smallest unused number that shares a factor with 6.
%t nn = 2^10; u = 1; c[_] = 0; MapIndexed[Set[{a[First[#2]], c[#1]}, {#1, First[#2]}], {1, 2}]; While[c[u] > 0, u++]; Do[m = Max[Array[a[i - #] &, 2]]; k = u; While[Or[c[k] > 0, CoprimeQ[m, k]], k++]; Set[{a[i], c[k]}, {k, i}]; If[k == u, While[c[u] > 0, u++]], {i, Length[s] + 1, nn}]; Array[a, nn] (* _Michael De Vlieger_, Apr 14 2022 *)
%Y Cf. A352968, A064413, A270139, A084937, A336957, A098550.
%K nonn
%O 1,2
%A _Scott R. Shannon_, Apr 13 2022
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