cp's OEIS Frontend

A384045 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused positive number that shares a factor with a(n-1) if it is greater than it, else it is coprime to a(n-1) if it is less than it.

%I A384045 #13 May 27 2025 10:34:43
%S A384045 1,2,4,3,6,5,10,7,14,9,8,12,11,22,13,26,15,18,17,16,20,19,38,21,24,23,
%T A384045 46,25,30,29,27,33,28,32,31,62,35,34,36,39,37,74,41,40,42,44,43,86,45,
%U A384045 48,47,94,49,56,51,50,52,54,53,106,55,60,59,57,63,58,64,61
%N A384045 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused positive number that shares a factor with a(n-1) if it is greater than it, else it is coprime to a(n-1) if it is less than it.
%C A384045 For the terms studied all primes appear in their natural order, and approximately 65% of all primes p are immediately followed by a term 2*p. These later terms form the upper of the two lines in the graph.
%C A384045 In the first 100000 terms the fixed points are 1, 2, 12, 18, 98, 182, 306, 380; it is likely no more exist.
%C A384045 The sequence is a permutation of the positive integers as the lowest unused number after k terms will always appear as it will eventually be coprime to a(j) for some j > k.
%H A384045 Scott R. Shannon, <a href="/A384045/b384045.txt">Table of n, a(n) for n = 1..10000</a>
%e A384045 a(3) = 4 as a(2) = 2 and 4 > 2 and shares a factor with it. Note 3 cannot be chosen as 3 > 2 but is coprime to 2.
%e A384045 a(4) = 3 as a(3) = 4 and 3 < 4 and is coprime to it.
%t A384045 nn = 120; c[_] := False; j = 2; u = 3; c[1] = c[2] = True;
%t A384045 {1, 2}~Join~Reap[Do[k = u;
%t A384045   While[And[k < j, Or[c[k], ! CoprimeQ[j, k]]], k++];
%t A384045     If[k >= j,
%t A384045       If[PrimePowerQ[j],
%t A384045         Set[{p, k}, {FactorInteger[j][[1, 1]], 1}]; While[c[k*p], k++]; k *= p,
%t A384045         While[Or[c[k], CoprimeQ[j, k]], k++] ] ];
%t A384045     Sow[k]; Set[{c[k], j}, {True, k}];
%t A384045     If[k == u, While[c[u], u++]],
%t A384045 {n, 3, nn}] ][[-1, 1]] (* _Michael De Vlieger_, May 27 2025 *)
%Y A384045 Cf. A375563, A375564, A373545, A373546, A064413.
%K A384045 nonn
%O A384045 1,2
%A A384045 _Scott R. Shannon_, May 18 2025