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.

1, 2, 3, 4, 8, 10, 13, 15, 16, 22, 23, 25, 30, 32, 36, 37, 38, 39, 41, 44, 46, 49, 50, 52, 53, 59, 60, 64, 69, 70, 71, 76, 78, 81, 82, 85, 88, 92, 97, 98, 104, 106, 109, 111, 115, 120, 125, 127, 128, 133, 134, 135, 136, 137, 140, 142, 145, 148, 149, 152, 156, 161, 162, 170, 176, 182

Offset

1,2

Comments

a(n) first differs from A190896(n-1) at n=10: a(10)=22, whereas A190896(9)=19.

Links

Table of n, a(n) for n=1..66.

Attila Kiss, Java code to generate terms.

Example

5 is not a term since its binary expansion is "101", which is the concatenation of earlier a(2)="10" and a(1)="1".
19 is not a term since its binary expansion is "10011", which is the concatenation of a(4)="100" and a(3)="11".

Mathematica

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 *)

Prog

(Python)
from itertools import islice
def agen(): # generator of terms
    an, bins, concats = 1, {"1"}, set()
    while True:
        yield an
        while (bn:=bin(an:=an+1)[2:]) in concats: pass
        concats |= {bn+bi for bi in bins} | {bi+bn for bi in bins}
        bins.add(bn)
print(list(islice(agen(), 62))) # Michael S. Branicky, Sep 29 2023

Crossrefs

Cf. A364871, A190896.

Sequence in context: A346971 A180941 A190896 * A135772 A209064 A222264

Adjacent sequences: A365015 A365016 A365017 * A365019 A365020 A365021

Keyword

nonn,base

Author

Attila Kiss, Aug 16 2023

Status

approved