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

Author

Attila Kiss, Aug 16 2023

Keywords

• nonn
• base

Comments

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

Examples

			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".
		

Links

• Attila Kiss, Java code to generate terms.

Crossrefs

Cf. A364871, A190896.

Programs

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 *)

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