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A365018 a(n) is the least positive integer not already in the sequence whose binary expansion is not the concatenation of any two earlier terms.

%I A365018 #51 Nov 05 2023 15:00:07
%S A365018 1,2,3,4,8,10,13,15,16,22,23,25,30,32,36,37,38,39,41,44,46,49,50,52,
%T A365018 53,59,60,64,69,70,71,76,78,81,82,85,88,92,97,98,104,106,109,111,115,
%U A365018 120,125,127,128,133,134,135,136,137,140,142,145,148,149,152,156,161,162,170,176,182
%N A365018 a(n) is the least positive integer not already in the sequence whose binary expansion is not the concatenation of any two earlier terms.
%C A365018 a(n) first differs from A190896(n-1) at n=10: a(10)=22, whereas A190896(9)=19.
%H A365018 Attila Kiss, <a href="/A365018/a365018.java.txt">Java code to generate terms</a>.
%e A365018 5 is not a term since its binary expansion is "101", which is the concatenation of earlier a(2)="10" and a(1)="1".
%e A365018 19 is not a term since its binary expansion is "10011", which is the concatenation of a(4)="100" and a(3)="11".
%t A365018 conc[x_, y_] := FromDigits[Flatten@IntegerDigits[{x, y}, 2], 2]; a[1] = 1; a[n_] := a[n] = Module[{k = a[n - 1] + 1, v = Array[a, n - 1], c}, c = conc @@@ Select[Tuples[v, {2}], UnsameQ @@ # &]; While[! FreeQ[c, k], k++]; k]; Array[a, 60] (* _Amiram Eldar_, Sep 29 2023 *)
%o A365018 (Python)
%o A365018 from itertools import islice
%o A365018 def agen(): # generator of terms
%o A365018     an, bins, concats = 1, {"1"}, set()
%o A365018     while True:
%o A365018         yield an
%o A365018         while (bn:=bin(an:=an+1)[2:]) in concats: pass
%o A365018         concats |= {bn+bi for bi in bins} | {bi+bn for bi in bins}
%o A365018         bins.add(bn)
%o A365018 print(list(islice(agen(),62))) # _Michael S. Branicky_, Sep 29 2023
%Y A365018 Cf. A364871, A190896.
%K A365018 nonn,base
%O A365018 1,2
%A A365018 _Attila Kiss_, Aug 16 2023