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A353905 a(1) = 1; for n > 1, a(n) = smallest positive number that has not appeared that has a common factor with a(n-1) + the smallest unseen positive number.

%I A353905 #21 Jan 16 2023 11:10:01
%S A353905 1,3,5,7,6,2,4,8,17,10,19,12,9,14,15,13,16,18,29,20,31,21,22,11,24,47,
%T A353905 25,26,28,27,30,53,32,33,34,36,59,38,61,35,40,39,42,45,44,67,46,23,48,
%U A353905 50,51,52,89,49,43,54,56,57,58,55,60,97,62,63,64,101,66,103,65,68,69,70,107,72,109
%N A353905 a(1) = 1; for n > 1, a(n) = smallest positive number that has not appeared that has a common factor with a(n-1) + the smallest unseen positive number.
%C A353905 The sequence is conjectured to be a permutation of the positive integers. In the first 100000 terms the number 18869 holds the record for the greatest number of terms for which it is the lowest unseen number, 4769 terms in all. In the same range there are fifteen fixed points, the last being a(1204), and it is likely no more exist.
%H A353905 Michael De Vlieger, <a href="/A353905/b353905.txt">Table of n, a(n) for n = 1..10000</a>
%H A353905 Michael De Vlieger, <a href="/A353905/a353905_1.png">Annotated log-log scatterplot of a(n)</a>, n = 1..2^14, showing records in red and local minima in blue. Primes and fixed points are highlighted in green and gold, respectively.
%H A353905 Scott R. Shannon, <a href="/A353905/a353905.png">Image of the first 100000 terms</a>. The green line is y = n.
%e A353905 a(2) = 3 as a(1) = 1, the smallest unseen positive number is 2, and 1 + 2 = 3, and 3 is the smallest number that has not yet appeared that shares a factor with 3.
%e A353905 a(5) = 6 as a(4) = 7, the smallest unseen positive number is 2, and 7 + 2 = 9, and 6 is the smallest number that has not yet appeared that shares a factor with 9.
%t A353905 nn = 120; a[1] = c[1] = 1; u = 2; Do[k = u; While[Nand[c[k] == 0, ! CoprimeQ[#, k]], k++] &[a[i - 1] + u]; Set[{a[i], c[k]}, {k, i}]; If[k == u, While[c[u] > 0, u++]], {i, 2, nn}]; Array[a, nn] (* _Michael De Vlieger_, May 15 2022 *)
%Y A353905 Cf. A337136, A089088, A064413, A085229, A093714.
%K A353905 nonn
%O A353905 1,2
%A A353905 _Scott R. Shannon_, May 10 2022