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%I #11 Jul 20 2018 11:55:05
%S 2,1,3,5,10,6,8,4,12,7,19,11,29,16,18,21,24,13,69,39,45,51,27,14,30,
%T 15,33,17,36,9,84,96,110,28,60,31,32,68,35,75,80,20,42,43,22,94,48,50,
%U 25,52,26,55,57,58,125,64,66,142,73,37,77,79,40,41,175,44
%N Lexicographically earliest sequence of distinct positive terms such that a(1) = 2 and for any n > 0, the binary representation of Sum_{k=1..n} a(k) starts with the binary representation of a(n).
%C This sequence is a binary variant of A316918.
%C This sequence is conjectured to be infinite.
%C This sequence is conjectured to be a permutation of the natural numbers.
%H Rémy Sigrist, <a href="/A316993/b316993.txt">Table of n, a(n) for n = 1..10000</a>
%H Rémy Sigrist, <a href="/A316993/a316993.png">Colored scatterplot of the first 200000 terms</a> (where the color is function of A070939(Sum_{k=1..n} a(k)) - A070939(a(n)))
%H Rémy Sigrist, <a href="/A316993/a316993_1.png">Scatterplot of the ordinal transform of the first 25000 terms of A070939(Sum_{k=1..n} a(k)) - A070939(a(n))</a>
%H Rémy Sigrist, <a href="/A316993/a316993.txt">C++ program for A316993</a>
%e The first terms, alongside the binary representations of a(n) and of Sum_{k=1..n} a(k), are:
%e n a(n) bin(a(n)) bin(Sum_{k=1..n} a(k))
%e -- ---- --------- ----------------------
%e 1 2 10 10
%e 2 1 1 11
%e 3 3 11 110
%e 4 5 101 1011
%e 5 10 1010 10101
%e 6 6 110 11011
%e 7 8 1000 100011
%e 8 4 100 100111
%e 9 12 1100 110011
%e 10 7 111 111010
%e 11 19 10011 1001101
%e 12 11 1011 1011000
%e 13 29 11101 1110101
%e 14 16 10000 10000101
%e 15 18 10010 10010111
%e 16 21 10101 10101100
%e 17 24 11000 11000100
%e 18 13 1101 11010001
%e 19 69 1000101 100010110
%e 20 39 100111 100111101
%o (C++) See Links section.
%Y Cf. A070939, A316918.
%K nonn,base
%O 1,1
%A _Rémy Sigrist_, Jul 18 2018