cp's OEIS Frontend

A354727 a(1) = 1; for n > 1, a(n) is the smallest positive number that has not yet appeared that shares a factor with a(n-1) and the difference a(n) - a(n-1) is distinct from all previous differences.

%I A354727 #33 Jun 05 2022 08:24:50
%S A354727 1,2,4,8,6,3,9,12,20,5,10,22,11,33,15,24,14,7,21,28,16,26,13,39,18,34,
%T A354727 17,51,27,42,36,32,50,25,45,40,64,30,55,35,56,48,75,66,38,19,57,76,44,
%U A354727 72,46,23,69,92,52,65,95,60,93,31,62,94,47,141,54,90,63,49,84,68,85,125,70,112,58
%N A354727 a(1) = 1; for n > 1, a(n) is the smallest positive number that has not yet appeared that shares a factor with a(n-1) and the difference a(n) - a(n-1) is distinct from all previous differences.
%C A354727 This sequences uses similar a similar rule to A354687 but here the sign of the difference between a(n-1) and a(n) is considered. The terms show an overall similar behavior to A354687 although here only two lines show a higher concentration of terms than the other lines. See the linked image.
%C A354727 In the first 100000 terms the fixed points are 1,2,15,32,100,115,300,720. It is plausible no more exist although this is unknown.. The sequence is conjectured to be a permutation of the positive integers.
%C A354727 See A354739 for the differences between terms.
%H A354727 Scott R. Shannon, <a href="/A354727/a354727.png">Image of the first 100000 terms</a>. The green line is y = n.
%e A354727 a(9) = 20 as a(8) = 12, and 20 is the smallest unused number that shares a factor with 12 and whose difference from the previous term, 20 - 12 = 8, has not appeared. Note that 10,14,15,16,18 all share a factor with 12 but their differences from 12, namely -2,2,3,4,6, have already appeared as differences between previous pairs of terms.
%Y A354727 Cf. A354739, A354687, A354575, A354679, A354688.
%K A354727 nonn
%O A354727 1,2
%A A354727 _Scott R. Shannon_, Jun 05 2022