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Lexicographically earliest sequence of distinct positive integers such that three consecutive terms are never pairwise coprime.
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%I #17 Jun 13 2021 03:24:09

%S 1,2,4,3,6,5,8,10,7,12,9,11,15,18,13,14,16,17,20,22,19,24,21,23,27,30,

%T 25,26,28,29,32,34,31,36,33,35,39,40,38,37,42,44,41,46,48,43,45,50,47,

%U 52,54,49,51,56,57,58,60,53,55,65,59,70,62,61,64,66,63

%N Lexicographically earliest sequence of distinct positive integers such that three consecutive terms are never pairwise coprime.

%C In other words, for any n > 0, at least one of gcd(a(n), a(n+1)), gcd(a(n), a(n+2)), gcd(a(n+1), a(n+2)) is strictly greater than 1.

%C This sequence has connections with the Yellowstone permutation (A098550).

%C Conjecture: this sequence is a permutation of the natural numbers.

%H Rémy Sigrist, <a href="/A338618/b338618.txt">Table of n, a(n) for n = 1..10000</a>

%H Rémy Sigrist, <a href="/A338618/a338618.gp.txt">PARI program for A338618</a>

%e The first terms, alongside associated GCD's, are:

%e n a(n) gcd(a(n),a(n+1)) gcd(a(n),a(n+2)) gcd(a(n+1),a(n+2))

%e -- ---- ---------------- ---------------- ------------------

%e 1 1 1 1 2

%e 2 2 2 1 1

%e 3 4 1 2 3

%e 4 3 3 1 1

%e 5 6 1 2 1

%e 6 5 1 5 2

%e 7 8 2 1 1

%e 8 10 1 2 1

%e 9 7 1 1 3

%e 10 12 3 1 1

%o (PARI) See Links section.

%Y See A338619 for a similar sequence.

%Y Cf. A084937, A098550.

%K nonn

%O 1,2

%A _Rémy Sigrist_, Nov 04 2020