cp's OEIS Frontend

A367742 Lexicographically earliest infinite sequence of distinct positive numbers such that, for n>3, a(n) has a common factor with a(n-2) and n but not with a(n-1).

%I A367742 #10 Dec 31 2023 19:33:25
%S A367742 1,2,3,4,15,8,21,10,9,5,33,20,39,14,27,16,51,22,57,26,45,28,69,32,75,
%T A367742 34,63,38,87,40,93,44,81,46,105,52,111,50,99,25,123,35,129,55,6,115,
%U A367742 94,135,56,65,12,13,106,117,80,91,18,203,118,145,122,155,183,62,165,58,201,64,141,68,213
%N A367742 Lexicographically earliest infinite sequence of distinct positive numbers such that, for n>3, a(n) has a common factor with a(n-2) and n but not with a(n-1).
%C A367742 This is a variation of the Yellowstone permutation A098550 with an additional restriction that each term a(n) must have a common factor with n. For the sequence to be infinite a(n) must be chosen so it does not have as factors all the prime factors of n+1. See the examples below.
%C A367742 Unlike A098550 the primes do not appear in their natural order, and in general are delayed in their appearance relative to similarly sized numbers. In the first 100000 terms the fixed points are 1, 2, 3, 4, 9, 14, 16, 74, 76, 86, 88, 207, 320, 322, 901; it is unknown if more exist. The sequence is conjectured to be a permutation of the positive numbers.
%H A367742 Scott R. Shannon, <a href="/A367742/b367742.txt">Table of n, a(n) for n = 1..10000</a>
%H A367742 Scott R. Shannon, <a href="/A367742/a367742.png">Image of the first 100000 terms for a(n) <= 500000</a>. The green line is a(n) = n.
%e A367742 a(5) = 15 as 15 shares a factor with a(3) = 3 and with n = 5, does not share a factor with a(4) = 4, and 15 does not have as factors all the prime factors of 5+1 = 6 = 2*3.
%e A367742 a(55) = 80 as 80 shares a factor with a(53) = 106 and with n = 55, does not share a factor with a(54) = 117, and 80 does not have as factors all the prime factors of 55+1 = 56 = 2^3*7. Note that 70 satisfies the first three criteria but not the last, so choosing a(55) = 70 would mean a(56) would not exist.
%Y A367742 Cf. A367741, A098550, A336957, A064413, A027748, A368231.
%K A367742 nonn
%O A367742 1,2
%A A367742 _Scott R. Shannon_, Nov 29 2023