A114801 2-concatenation-free sequence starting (1,2).

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 20, 22, 30, 33, 40, 44, 50, 55, 60, 66, 70, 77, 80, 88, 90, 99, 100, 121, 123, 124, 125, 126, 127, 128, 129, 131, 132, 134, 135, 136, 137, 138, 139, 141, 142, 143, 145, 146, 147, 148, 149, 151, 152, 153, 154, 156, 157, 158

Offset

1,2

Comments

Starting with the terms (1,2) this sequence consists of minimum increasing terms such that no term is the concatenation of any two previous distinct terms. The next consecutive number skipped after 121 is 122 = Concatenate(1, 22). This is the analog of a 2-Stöhr sequence with concatenation (base 10) substituting for addition. A033627 "0-additive sequence: not the sum of any previous pair" is another name for the 2-Stöhr sequence.

Links

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Stöhr Sequence.

Rémy Sigrist, PARI program for A114801

Formula

a(0) = 1, a(1) = 2, for n>2: a(n) = least k > a(n-1) such that k is not an element of {Concatenate(a(i), a(j))} for any distinct a(i) <= a(n-1) and a(j) <= a(n-1).

Mathematica

conc[x_, y_] := FromDigits@ Flatten@ IntegerDigits[{x, y}]; L = {1, 2}; cc = {12, 21}; Do[k = 1 + Max@L; While[MemberQ[cc, k], k++]; cc = Union[cc, conc[#, k] & /@ L, conc[k, #] & /@ L]; AppendTo[L, k]; , {65}]; L (* Giovanni Resta, Jun 15 2016 *)

Prog

(PARI) See Links section.
(Python)
from itertools import count, islice
def agen(): # generator of terms
    cats1, cats2, an, s = {"1", "2"}, {"12", "21"}, 3, "3"
    yield from [1, 2]
    while True:
        yield an
        cats2 |= {s + c for c in cats1} | {c + s for c in cats1}
        cats1.add(s)
        while (s:=str(an)) in cats1 or s in cats2:
            an += 1
print(list(islice(agen(), 59))) # Michael S. Branicky, Feb 01 2024

Crossrefs

Cf. A084383, A033627.

Sequence in context: A125289 A180482 A193460 * A114802 A055933 A188650

Adjacent sequences: A114798 A114799 A114800 * A114802 A114803 A114804

Keyword

base,easy,nonn

Author

Jonathan Vos Post, Feb 18 2006

Extensions

Data corrected by Giovanni Resta, Jun 14 2016

Status

approved