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Lexicographically first sequence of distinct terms, beginning with a(1)=5, with the property that each triple of consecutive terms contains a term that divides the sum of the other two terms.
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%I #16 Jan 22 2017 21:35:40

%S 5,1,2,3,4,7,9,8,10,6,14,16,12,20,28,24,13,11,15,18,21,33,27,30,19,41,

%T 22,25,47,36,58,50,42,23,61,31,32,63,65,64,43,85,44,91,45,17,40,57,74,

%U 97,51,37,60,83,106,129,152,175,109,66,35,39,82,113,133,93

%N Lexicographically first sequence of distinct terms, beginning with a(1)=5, with the property that each triple of consecutive terms contains a term that divides the sum of the other two terms.

%C The initial term a(1)=5 seems to be the least one that leads to a sequence that is not ultimately linear.

%C The variant with:

%C - a(1)=1 matches A000027,

%C - a(1)=2 matches A181440,

%C - a(1)=3 starts with 3, 1, 2, and then matches A000027,

%C - a(1)=4 starts with 4, 1, 2, and then matches A143097,

%C - a(1)=6 starts with 6, 1, 2, 3, 4, 5, and then matches A000027,

%C - a(1)=9 starts with 9, 1, 2, 3, 4, 5, 6, 7, 8, 13, 11, 12, 10, and then matches A143097.

%C Conjecturally, all other variants are not ultimately linear.

%H Rémy Sigrist, <a href="/A281409/b281409.txt">Table of n, a(n) for n = 1..10000</a>

%H Rémy Sigrist, <a href="/A281409/a281409.gp.txt">PARI program for A281409</a>

%e The first terms, alongside the indexes of the terms that divide the sum of the other two terms within the n-th triple of consecutive terms, are:

%e n a(n) Indexes

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

%e 1 5 2, 3

%e 2 1 1, 2, 3

%e 3 2 2

%e 4 3 3

%e 5 4 1

%e 6 7 3

%e 7 9 1

%e 8 8 1, 3

%e 9 10 1, 2

%e 10 6 1

%e 11 14 1

%e 12 16 1, 2

%e 13 12 1, 2

%e 14 20 3

%e 15 28 3

%e 16 24 1

%e 17 13 1

%e 18 11 1

%e 19 15 2

%e 20 18 1

%e 21 21 3

%e 22 33 3

%e 23 27 3

%e 24 30 1

%e 25 19 2

%Y Cf. A000027, A143097, A181440, A278962, A281408.

%K nonn

%O 1,1

%A _Rémy Sigrist_, Jan 21 2017