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Lexicographically earliest sequence of positive integers such that the pairs (gcd(a(n), a(n+1)), a(n+1)) are all distinct.
2

%I #10 Aug 04 2024 12:42:52

%S 1,1,2,2,3,3,4,4,5,5,6,4,6,6,7,7,8,8,9,6,8,10,10,11,10,12,8,12,9,9,12,

%T 12,13,12,14,14,15,10,15,14,16,16,17,16,18,12,15,15,18,18,19,18,20,16,

%U 20,20,21,14,21,20,22,11,22,22,23,22,24,16,24,18,21

%N Lexicographically earliest sequence of positive integers such that the pairs (gcd(a(n), a(n+1)), a(n+1)) are all distinct.

%C The value 1 appears twice; for any v > 1, the value v appears A000005(v) times.

%H Rémy Sigrist, <a href="/A375114/b375114.txt">Table of n, a(n) for n = 1..10000</a>

%H Rémy Sigrist, <a href="/A375114/a375114.gp.txt">PARI program</a>

%e The first terms, alongside the corresponding pairs, are:

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

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

%e 1 1 (1, 1)

%e 2 1 (1, 2)

%e 3 2 (2, 2)

%e 4 2 (1, 3)

%e 5 3 (3, 3)

%e 6 3 (1, 4)

%e 7 4 (4, 4)

%e 8 4 (1, 5)

%e 9 5 (5, 5)

%e 10 5 (1, 6)

%e 11 6 (2, 4)

%e 12 4 (2, 6)

%e 13 6 (6, 6)

%e 14 6 (1, 7)

%e 15 7 (7, 7)

%o (PARI) \\ See Links section.

%Y See A375110 for similar sequences.

%Y Cf. A000005.

%K nonn

%O 1,3

%A _Rémy Sigrist_, Jul 30 2024