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Lexicographically earliest infinite sequence of distinct positive terms such that for any n > 1, the binary representation of a(n) appears as a substring in the binary representation of Sum_{k=1..n-1} a(k).
2

%I #11 Aug 19 2018 02:47:26

%S 2,1,3,6,4,8,12,9,5,18,17,10,7,19,14,16,11,20,13,24,22,15,32,36,34,25,

%T 23,37,27,21,26,64,69,40,43,29,30,35,39,44,28,42,53,129,72,38,31,81,

%U 45,50,46,47,49,74,41,54,55,51,52,57,58,128,68,70,140,77,60

%N Lexicographically earliest infinite sequence of distinct positive terms such that for any n > 1, the binary representation of a(n) appears as a substring in the binary representation of Sum_{k=1..n-1} a(k).

%C The sequence must start with a(1) = 2 in order to be infinite, and for any n > 1, a(n) <= Sum_{k=1..n-1} a(k).

%C This sequence has similarities with A160855.

%H Rémy Sigrist, <a href="/A317788/b317788.txt">Table of n, a(n) for n = 1..10000</a>

%H Rémy Sigrist, <a href="/A317788/a317788.png">Density plot of the first 100000000 terms</a>

%H Rémy Sigrist, <a href="/A317788/a317788.txt">C++ program for A317788</a>

%e The first terms, alongside the binary representations of a(n) and of Sum_{k=1..n-1} a(k), are:

%e n a(n) bin(a(n)) bin(Sum_{k=1..n-1} a(k))

%e -- ---- --------- ------------------------

%e 1 2 10 0

%e 2 1 1 10

%e 3 3 11 11

%e 4 6 110 110

%e 5 4 100 1100

%e 6 8 1000 10000

%e 7 12 1100 11000

%e 8 9 1001 100100

%e 9 5 101 101101

%e 10 18 10010 110010

%o (C++) See Links section.

%Y Cf. A160855.

%K nonn,base

%O 1,1

%A _Rémy Sigrist_, Aug 07 2018