%I #8 Oct 01 2024 08:46:38
%S 1,2,1,3,1,4,1,2,1,5,1,6,1,2,1,3,1,7,1,8,1,2,1,4,1,2,1,9,1,3,1,10,1,2,
%T 1,5,1,11,1,12,1,2,1,3,1,4,1,2,1,6,1,2,1,3,1,13,1,14,1,2,1,7,1,15,1,3,
%U 1,5,1,16,1,2,1,4,1,2,1,8,1,2,1,4,1,2,1
%N Lexicographically earliest sequence of positive integers such that for any n > 0, there is an even number of k's such that 1 <= k < n and a(n) divides a(k).
%C The sequence is well defined as we can always extend it with a number greater than any prior term.
%H Rémy Sigrist, <a href="/A376566/b376566.txt">Table of n, a(n) for n = 1..10000</a>
%F a(2*k-1) = 1 for any k > 0.
%F a(2*A025523(n-1)) = n for any n > 1 (and this is the first occurrence of n in the sequence). - _Hugo Pfoertner_, Oct 01 2024
%e The first terms, alongside the corresponding k's, are:
%e n a(n) k's
%e -- ---- ---------------------------------------------
%e 1 1 None
%e 2 2 None
%e 3 1 1, 2
%e 4 3 None
%e 5 1 1, 2, 3, 4
%e 6 4 None
%e 7 1 1, 2, 3, 4, 5, 6
%e 8 2 2, 6
%e 9 1 1, 2, 3, 4, 5, 6, 7, 8
%e 10 5 None
%e 11 1 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
%e 12 6 None
%e 13 1 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
%e 14 2 2, 6, 8, 12
%e 15 1 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14
%e 16 3 4, 12
%o (PARI) { nb = vector(16); for (n = 1, 86, for (v = 1, oo, if (nb[v]%2==0, print1 (v ", "); fordiv (v, d, nb[d]++;); break;););); }
%Y Cf. A025523.
%K nonn
%O 1,2
%A _Rémy Sigrist_, Sep 28 2024