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%I #12 Dec 20 2017 12:13:33
%S 1,2,4,3,6,7,14,8,16,9,18,5,10,11,21,12,22,13,23,27,24,28,26,29,53,31,
%T 54,32,56,17,57,35,15,36,61,37,63,19,64,39,33,20,34,41,69,42,71,43,72,
%U 44,73,45,74,46,38,47,77,48,78,49,79,25,40,51,81,52,82
%N Lexicographically earliest sequence of distinct positive terms such that, for any n > 0, the binary expansion of a(n) * a(n + 1) starts with the binary expansion of n.
%C It is likely that this sequence is a permutation of the natural numbers.
%C The lines visible in the scatterplot of the first terms seems to corresponds to set of indices n where the function f(n) = Sum_{k=1..n-1} (-1)^k * (A029837(1+a(k)*a(k+1)) - A029837(1+k)) has the same value; those lines can be partitioned into two groups, depending on the parity of n (see Links section).
%C This sequence has connections with A272679: here the binary expansion of a(n)*a(n+1) starts with that of n, there the binary expansion of a(n)^2 starts with that of n.
%H Rémy Sigrist, <a href="/A296616/b296616.txt">Table of n, a(n) for n = 1..10000</a>
%H Rémy Sigrist, <a href="/A296616/a296616.txt">C++ program for A296616</a>
%H Rémy Sigrist, <a href="/A296616/a296616.png">Colored scatterplot of the first 100000 terms</a> (where the color is function of Sum_{k=1..n-1} (-1)^k * (A029837(1+a(k)*a(k+1)) - A029837(1+k)))
%H Rémy Sigrist, <a href="/A296616/a296616_1.png">Colored scatterplot of the first 10000 terms</a> (where the color is function of the parity of n)
%e The first terms, alongside the binary representations of n and a(n) * a(n + 1), are:
%e n a(n) bin(n) bin(a(n)*a(n+1))
%e -- ---- ------ ----------------
%e 1 1 1 10
%e 2 2 10 1000
%e 3 4 11 1100
%e 4 3 100 10010
%e 5 6 101 101010
%e 6 7 110 1100010
%e 7 14 111 1110000
%e 8 8 1000 10000000
%e 9 16 1001 10010000
%e 10 9 1010 10100010
%e 11 18 1011 1011010
%e 12 5 1100 110010
%e 13 10 1101 1101110
%e 14 11 1110 11100111
%e 15 21 1111 11111100
%e 16 12 10000 100001000
%e 17 22 10001 100011110
%e 18 13 10010 100101011
%e 19 23 10011 1001101101
%e 20 27 10100 1010001000
%o (C++) See Links section.
%Y Cf. A029837, A272679.
%K nonn,base
%O 1,2
%A _Rémy Sigrist_, Dec 17 2017
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