A386551 Numbers that can terminate in 1 by successively dividing a contiguous substring of digits by 2.

1, 2, 4, 8, 16, 26, 32, 34, 38, 46, 52, 54, 58, 62, 64, 68, 76, 86, 92, 94, 98, 102, 104, 108, 112, 114, 116, 118, 122, 124, 126, 128, 132, 134, 136, 138, 142, 144, 146, 148, 152, 154, 156, 158, 162, 164, 166, 168, 172, 174, 176, 178, 182, 184, 186, 188, 192, 194

Offset

1,2

Comments

As in A389140, the substring replaced must be even and cannot begin with a 0 digit, and there may be multiple different replacements possible and the aim is to find whether any sequence of steps can reach final value 1.

All even numbers, except multiples of 5 and the 24 even numbers that terminate at 3 (see A389305), appear in this sequence.

No odd terms > 1. - Michael S. Branicky, Sep 28 2025

Links

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Example

176 is a term since it can reach 1 by trajectory 176-->138-->134-->132-->116-->58-->54-->52-->26-->16-->8-->4-->2-->1.

Prog

(Python)
from functools import cache
def children(n):
    s = str(n)
    return set(int(s[:i]+str(q>>1)+s[j:]) for i in range(len(s)) for j in range(i+1, len(s)+1) if (w:=s[i:j])[0]!='0' and (q:=int(w))&1==0)
@cache
def ok(n):
    if n > 1 and n&1: return False # no odds other than 1
    if n.bit_count() == 1: return True # powers of 2
    return any(ok(c) for c in children(n))
print([k for k in range(195) if ok(k)]) # Michael S. Branicky, Sep 28 2025

Crossrefs

Supersequence of A000079.

Cf. A389140, A389305, A389141.

Sequence in context: A070789 A302588 A382381 * A348656 A354146 A354255

Adjacent sequences: A386548 A386549 A386550 * A386552 A386553 A386554

Keyword

nonn,base

Author

Ali Sada, Sep 27 2025

Status

approved