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A349472 a(1)=1; for n > 1, a(n) is the smallest unused positive number such that gcd(a(n-1)+n,a(n)) > 1.

%I A349472 #18 Oct 06 2022 08:17:29
%S A349472 1,3,2,4,6,8,5,13,10,12,23,7,14,16,31,47,18,9,20,15,21,43,11,25,22,24,
%T A349472 17,27,26,28,59,35,30,32,67,103,34,33,36,19,38,40,83,127,42,44,39,29,
%U A349472 45,50,101,48,202,46,303,359,52,54,113,173,51,226,68,55,56,58,60,62,131,57,64,66
%N A349472 a(1)=1; for n > 1, a(n) is the smallest unused positive number such that gcd(a(n-1)+n,a(n)) > 1.
%C A349472 After 200000 terms the smallest unused number is 28349, although like similar sequences this is almost certainly a permutation of the positive integers. In the same range the fixed points are 4, 21, 50, 1269; it is likely no more exist. See the linked image.
%H A349472 Michael De Vlieger, <a href="/A349472/b349472.txt">Table of n, a(n) for n = 1..10000</a>
%H A349472 Scott R. Shannon, <a href="/A349472/a349472.png">Image of the first 200000 terms</a>. The green line is y = n.
%e A349472 a(2) = 3 as a(1) + 2 = 3, 3 has not previously appeared, and gcd(3,3) > 1.
%e A349472 a(3) = 2 as a(2) + 3 = 6, 2 has not previously appeared, and gcd(6,2) > 1.
%e A349472 a(12) = 7 as a(11) + 12 = 35, 7 has not previously appeared, and gcd(35,7) > 1.
%t A349472 nn = 72; c[_] = False; u = 2; a[1] = j = 1; c[1] = True; Do[Set[{k, m}, {u, n + j}]; While[Or[c[k], CoprimeQ[k, m]], k++]; Set[{a[n], c[k], j}, {k, True, k}]; If[k == u, While[c[u], u++]], {n, 2, nn}]; Array[a, nn] (* _Michael De Vlieger_, Oct 05 2022 *)
%o A349472 (Python)
%o A349472 from math import gcd
%o A349472 terms, appears = [1], {}
%o A349472 for n in range(2, 100):
%o A349472     t = 2
%o A349472     while True:
%o A349472         if appears.get(t) is None and gcd(terms[-1]+n, t) > 1  :
%o A349472             appears[t] = True; terms.append(t); break
%o A349472         t += 1
%o A349472 print(terms) # _Gleb Ivanov_, Nov 19 2021
%Y A349472 Cf. A064413, A347113, A098550, A336957.
%K A349472 nonn
%O A349472 1,2
%A A349472 _Scott R. Shannon_, Nov 19 2021