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a(1) = 1, and for any n > 0, a(2*n) = a(n), a(2*n+1) is the least positive number not yet in the sequence coprime to a(n) and a(n+1).
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%I #12 Dec 16 2023 13:10:45

%S 1,1,2,1,3,2,5,1,4,3,7,2,9,5,6,1,11,4,13,3,8,7,15,2,17,9,14,5,19,6,23,

%T 1,10,11,21,4,25,13,16,3,29,8,27,7,22,15,31,2,33,17,20,9,37,14,39,5,

%U 12,19,35,6,41,23,18,1,43,10,47,11,26,21,53,4,49,25

%N a(1) = 1, and for any n > 0, a(2*n) = a(n), a(2*n+1) is the least positive number not yet in the sequence coprime to a(n) and a(n+1).

%C This sequence has similarities with Stern's diatomic series (A002487) as it equals its even bisection and two consecutive terms are always coprime.

%C For any prime number p, p is the first multiple of p to appear in the sequence.

%C All prime numbers appear in the sequence, in natural order.

%C Will every positive integer appear?

%H Rémy Sigrist, <a href="/A368071/b368071.txt">Table of n, a(n) for n = 1..8192</a>

%H Rémy Sigrist, <a href="/A368071/a368071.gp.txt">PARI program</a>

%F gcd(a(n), a(n+1)) = 1.

%e a(1) = 1 by definition.

%e a(2) = a(1) = 1.

%e a(3) must be coprime to a(1) = 1 and a(2) = 1; we chose a(3) = 2.

%e a(4) = a(2) = 1.

%e a(5) must be coprime to a(2) = 1 and a(3) = 2; we chose a(5) = 3.

%e a(6) = a(3) = 2.

%e a(7) must be coprime to a(3) = 2 and a(4) = 1; we chose a(7) = 5.

%o (PARI) See Links section.

%Y Cf. A002487, A368147.

%K nonn

%O 1,3

%A _Rémy Sigrist_, Dec 10 2023