%I #17 Oct 25 2022 13:48:32
%S 1,2,5,3,7,4,9,8,15,11,6,17,10,19,13,21,23,12,25,16,31,14,33,20,37,18,
%T 41,26,47,22,49,27,43,29,35,53,24,55,28,57,34,59,38,71,30,67,32,73,36,
%U 79,39,61,45,77,46,81,91,40,87,44,83,50,99,52,97,42,95,51,65,89,63,101,48,103,54,113
%N a(1) = 1; for n > 1, a(n) is the smallest positive number that has not yet appeared that is coprime to a(n-1) and the difference a(n) - a(n-1) is distinct from all previous differences.
%C This sequence uses a similar rule to A354688 but here the sign of the difference between a(n-1) and a(n) is considered. This leads to the terms showing much more erratic behavior than A354688; see the linked image.
%C In the first 200000 terms the fixed points are 1,2,8,35, and it is likely no more exist. The sequence is conjectured to be a permutation of the positive integers.
%C See A354679 for the differences between terms.
%H Scott R. Shannon, <a href="/A354575/a354575.png">Image of the first 200000 terms</a>. The green line is y = n.
%e a(9) = 15 as a(8) = 8, and 15 is the smallest unused number that is coprime to 8 and whose difference from the previous term, 15 - 8 = 7, has not appeared. Note that 11 and 13 are coprime to 8 but their differences from 8, namely 3 and 5, have already appeared as differences between previous pairs of terms.
%e a(15) = 13 as a(14) = 19, and 13 is the smallest unused number that is coprime to 19 and whose difference from the previous term, 13 - 19 = -6, has not appeared. Note that 12 is coprime to 19 and smaller than 13 but its difference from 19, namely -7, has already appeared as a difference between a(13) and a(12).
%Y Cf. A354679, A354688, A354727, A354739, A354687.
%K nonn,look
%O 1,2
%A _Scott R. Shannon_, Jun 05 2022