cp's OEIS Frontend

A378116 Lexicographically earliest sequence of distinct positive integers such that a(a(n)) shares a factor with a(a(n)-2) while not sharing a factor with a(a(n)-1).

%I A378116 #9 Nov 30 2024 08:50:52
%S A378116 3,4,9,8,7,6,35,12,25,11,15,22,14,33,16,21,18,49,20,63,26,27,19,24,95,
%T A378116 28,45,32,31,30,217,34,77,36,55,38,39,40,51,44,42,121,46,99,50,57,43,
%U A378116 48,215,52,75,56,54,91,58,65,62,85,60,119,64,105,68,69,70,61,71,122,213,74,81,73,78,365,76,115,82,125,83,80,249,86,87,88,93,92,111,94,84
%N A378116 Lexicographically earliest sequence of distinct positive integers such that a(a(n)) shares a factor with a(a(n)-2) while not sharing a factor with a(a(n)-1).
%C A378116 A self-referencing Yellowstone permutation-like sequence. Care must be taken when finding a(n) to ensure that, if n+1 has previously appeared in the sequence, a(n-1) has at least one prime factor not in a(n), else a(n+1) would not exist.
%C A378116 In the first 100000 terms the fixed points are 6, 24, 30, 48, 80, 240, 629, 2328, 2532, 3960, 17130, 29850, 57480, 57876, 60180. It is likely more exist.
%C A378116 The missing numbers are 1, 2, 5, 10, 13, 17, 23, 29, 37, 41, 47, 53, 59, 66 ... .
%H A378116 Scott R. Shannon, <a href="/A378116/b378116.txt">Table of n, a(n) for n = 1..10000</a>
%H A378116 Scott R. Shannon, <a href="/A378116/a378116.png">Image of the first 100000 terms</a>. The green line is a(n) = n.
%e A378116 a(1) = 3 as a(3) = 9 and 9 is the smallest unused number that shares a factor with a(3-2) = a(1) = 3 while not sharing a factor with a(3-1) = a(2) = 4.
%e A378116 a(5) = 7 as a(7) = 35 and 35 is the smallest unused number that shares a factor with a(7-2) = a(5) = 7 while not sharing a factor with a(7-1) = a(6) = 6. Note that a(5) cannot be 5 as 5 does not share a factor with a(5-2) = a(3) = 9, nor can it be 6 as that would imply a(6) shares a factor with a(6-2) = a(4) = 8 while not sharing a factor with a(6-1) = a(5) = 6, which is impossible.
%Y A378116 Cf. A098550, A378098, A378030, A064413, A378030, A121053, A027746.
%K A378116 nonn
%O A378116 1,1
%A A378116 _Scott R. Shannon_, Nov 17 2024