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A333905 Lexicographically earliest sequence of distinct positive integers such that a(n) divides the concatenation of a(n+1) to a(n+2).

%I A333905 #14 Oct 26 2021 21:33:28
%S A333905 1,2,3,4,5,6,10,8,20,16,40,32,80,64,160,128,320,256,640,512,1280,1024,
%T A333905 2560,2048,5120,4096,10240,8192,20480,16384,40960,32768,81920,65536,
%U A333905 163840,131072,327680,262144,655360,524288,1310720,1048576,2621440,2097152,5242880,4194304,10485760,8388608,20971520,16777216,41943040
%N A333905 Lexicographically earliest sequence of distinct positive integers such that a(n) divides the concatenation of a(n+1) to a(n+2).
%F A333905 Conjectures from _Colin Barker_, Apr 09 2020: (Start)
%F A333905 G.f.: x*(1 + 2*x + x^2 - x^4 - 2*x^5 - 4*x^7) / (1 - 2*x^2).
%F A333905 a(n) = 2*a(n-2) for n>6.
%F A333905 (End)
%F A333905 Conjecture: a(n) = 2^((n-7)/2)*(5 + 2*sqrt(2) + (2*sqrt(2) - 5)*(-1)^n) for n > 6. - _Stefano Spezia_, Oct 23 2021
%e A333905 a(1) = 1 divides 23 (and 23 is a(2) = 2 concatenated to a(3) = 3);
%e A333905 a(2) = 2 divides 34 (and 34 is a(3) = 3 concatenated to a(4) = 4);
%e A333905 a(3) = 3 divides 45 (and 45 is a(4) = 4 concatenated to a(5) = 5);
%e A333905 a(4) = 4 divides 56 (and 56 is a(5) = 5 concatenated to a(6) = 6);
%e A333905 a(5) = 5 divides 610 (and 610 is a(6) = 6 concatenated to a(7) = 10);
%e A333905 a(6) = 6 divides 108 (and 108 is a(7) = 10 concatenated to a(8) = 8);
%e A333905 From a(7) = 10 on, the pattern of the sequence is regular.
%Y A333905 Cf. A085946 (a(1) = 1, a(2) = 2 and a(n) = smallest number not included earlier that divides the concatenation a(n-2), a(n-1)).
%K A333905 base,nonn
%O A333905 1,2
%A A333905 _Eric Angelini_, Apr 09 2020