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A306696 Lexicographically earliest sequence of nonnegative terms such that for any n > 0 and k > 0, if a(n) >= a(n+k), then a(n+2*k) <> a(n+k).

%I A306696 #13 Mar 09 2019 00:24:16
%S A306696 0,0,1,0,1,1,2,0,2,1,3,1,2,2,3,0,3,2,4,1,3,3,4,1,4,2,5,2,4,3,5,0,5,3,
%T A306696 6,2,4,4,6,1,5,3,7,3,5,4,6,1,6,4,7,2,5,5,7,2,6,4,8,3,6,5,7,0,7,5,8,3,
%U A306696 6,6,8,2,7,4,9,4,7,6,8,1,8,5,9,3,7,7,9
%N A306696 Lexicographically earliest sequence of nonnegative terms such that for any n > 0 and k > 0, if a(n) >= a(n+k), then a(n+2*k) <> a(n+k).
%C A306696 This sequence has graphical features in common with A286326.
%H A306696 Rémy Sigrist, <a href="/A306696/b306696.txt">Table of n, a(n) for n = 1..10000</a>
%H A306696 Rémy Sigrist, <a href="/A306696/a306696.txt">C++ program for A306696</a>
%H A306696 Rémy Sigrist, <a href="/A306696/a306696.png">Colored scatterplot of (n, a(n)) for n = 1..100000</a> (where the color is function of the least k >= 0 such that (1+a(n))/n >= 2^k/A007583(k))
%F A306696 Empirically:
%F A306696 - a(n) = 0 iff n is a power of 2 (A000079),
%F A306696 - a(n) = 1 iff n = 3 or belongs to A164095,
%F A306696 - a(2*n) = a(n),
%F A306696 - A181497(n) is the least k such that a(k) = n.
%e A306696 For n=1:
%e A306696 - a(1) = 0 is suitable.
%e A306696 For n=2:
%e A306696 - a(2) = 0 is suitable.
%e A306696 For n=3:
%e A306696 - a(1) = 0 >= a(2) = 0, so a(3) <> 0,
%e A306696 - a(3) = 1 is suitable.
%e A306696 For n=4:
%e A306696 - a(2) = 0 < a(3) = 1,
%e A306696 - a(4) = 0 is suitable.
%e A306696 For n=5:
%e A306696 - a(3) = 1 >= a(4) = 0, so a(5) <> 0,
%e A306696 - a(1) = 0 < a(3) = 1,
%e A306696 - a(5) = 1 is suitable.
%e A306696 For n=6:
%e A306696 - a(4) = 0 < a(5) = 1,
%e A306696 - a(2) = 0 >= a(4) = 0, so a(6) <> 0,
%e A306696 - a(6) = 1 is suitable.
%e A306696 For n=7:
%e A306696 - a(5) = 1 >= a(6) = 1, so a(7) <> 1,
%e A306696 - a(3) = 1 >= a(5) = 1, so a(7) <> 1,
%e A306696 - a(1) = 0 >= a(4) = 0, so a(7) <> 0,
%e A306696 - a(7) = 2 is suitable.
%o A306696 (C++) See Links section.
%Y A306696 Cf. A000079, A007583, A164095, A181497, A286326.
%K A306696 nonn
%O A306696 1,7
%A A306696 _Rémy Sigrist_, Mar 05 2019