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A288118 Lexicographically earliest sequence of distinct positive terms such that, for any i and j > 0, a(i*j) != a(i) * a(j).

%I A288118 #8 Jun 26 2017 07:58:58
%S A288118 2,1,3,4,5,6,7,8,10,9,11,13,12,14,16,15,17,19,18,21,20,22,23,25,24,26,
%T A288118 27,29,28,31,30,33,32,34,36,35,37,38,39,41,40,43,42,45,44,46,47,49,48,
%U A288118 50,52,51,53,54,56,55,57,58,59,60,61,62,63,65,64,67,66
%N A288118 Lexicographically earliest sequence of distinct positive terms such that, for any i and j > 0, a(i*j) != a(i) * a(j).
%C A288118 If we drop the unicity constraint, then we obtain the Liouville's function (A007421).
%C A288118 This sequence is a permutation of the natural numbers (we can always choose the least value not yet seen at prime positions).
%C A288118 Conjecturally:
%C A288118 - the sequence is self-inverse,
%C A288118 - | a(n) - n | <= 1 for any n > 0,
%C A288118 - | a(i)*a(j) - i*j | <> 1 for any i > 0 and j > 0,
%C A288118 - a(n) = n+1 iff a(n+1) = n.
%C A288118 a(6) = a(1) * a(2) * a(3).
%C A288118 This sequence has connections with A288119; here we avoid a(i)*a(j) = a(i*j), there a(i)+a(j) = a(i*j).
%H A288118 Rémy Sigrist, <a href="/A288118/b288118.txt">Table of n, a(n) for n = 1..10000</a>
%e A288118 a(1) cannot equal 1 as a(1*1) != a(1)*a(1); a(1) = 2 is acceptable.
%e A288118 a(2) cannot equal a(1); a(2) = 1 is acceptable.
%e A288118 a(3) cannot equal a(1), a(2); a(3) = 3 is acceptable.
%e A288118 a(4) cannot equal a(1)...a(3), a(2)^2; a(4) = 5 is acceptable.
%e A288118 a(5) cannot equal a(1)...a(4); a(5) = 4 is acceptable.
%e A288118 a(6) cannot equal a(1)...a(5), a(2)*a(3); a(6) = 6 is acceptable.
%Y A288118 Cf. A007421, A288119.
%K A288118 nonn
%O A288118 1,1
%A A288118 _Rémy Sigrist_, Jun 05 2017