A367618 -id:A367618 - cp's OEIS frontend

A367619 a(n) is the most remote positive ancestor of n in the comma-child graph in base 3.

1, 2, 3, 3, 1, 1, 7, 1, 2, 2, 7, 1, 1, 2, 1, 1, 1, 1, 7, 1, 1, 2, 1, 7, 1, 7, 1, 1, 1, 1, 1, 1, 7, 7, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 7, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 7, 1, 7, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 7, 1, 7, 1, 1, 1, 1, 1, 1, 7

Offset: 1

Michael S. Branicky and N. J. A. Sloane, Dec 20 2023

Analogous to A367617, but the calculations are done in base 3.
See A367338 for definitions of comma-child.
The sequence consists entirely of terms in {1, 2, 3, 7}.  In particular, two terms, a(3) = a(4) = 3; five terms, a(2,9,10,14,22) = 2; and 490 terms are 7, ending with a(2182). All other terms a(k) are 1, since a(2183..2190) = 1 and 1 <= p(n) - n <= b^2 - 1 (= 8 for base b = 3).

Michael S. Branicky, Table of n, a(n) for n = 1..2200
Eric Angelini, Michael S. Branicky, Giovanni Resta, N. J. A. Sloane, and David W. Wilson, The Comma Sequence: A Simple Sequence With Bizarre Properties, arXiv:2401.14346, Youtube

Cf. A367338, A367355, A367617, A367618.

from functools import cache
from sympy.ntheory.factor_ import digits
def comma_parent(n, base=3): # A367618(n)
    y = digits(n, base)[1]
    x = (n-y)%base
    k = n - y - base*x
    return k if k > 0 else -1
@cache
def a(n):
    cp = comma_parent(n)
    if cp <= 0: return n
    return a(cp)
print([a(n) for n in range(1, 88)])

a(n) is defined as n if A367618(n) = -1, else A367618(A367618(n)).

A367621 The lexicographically earliest infinite sequence of positive numbers in which each term is a comma-child of the previous term in base 3.

Original entry on oeis.org

1, 5, 12, 13, 18, 20, 27, 28, 32, 39, 40, 44, 51, 52, 57, 59, 67, 72, 74, 81, 82, 86, 93, 94, 98, 105, 106, 110, 117, 118, 122, 129, 130, 134, 141, 142, 146, 153, 154, 158, 166, 171, 173, 181, 186, 188, 196, 201, 203, 211, 216, 218, 226, 231, 233, 241, 245, 252

Offset: 1

Michael S. Branicky and N. J. A. Sloane, Dec 20 2023

Analogous to A367620, but with comma-children computed in base 3 (terms are shown in base 10, however).
We know from A367619 that the comma-child graph in base 3, starting at 1, is an infinite tree rooted at 1. By König's Infinity Lemma, an infinite path in that graph exists and hence this sequence is well defined for all n.  Therefore, at any bifurcation point, one or both forks will extend to infinity. The definition of this sequence requires that we choose the smallest fork that has an infinite continuation.
The terms in the data and b-file include a number of bifurcation points, but in each case the path chosen was the only one that did not lead to a finite sequence; see linked a-file.
We conjecture that choosing down-up-down-up-... is an infinite path, visiting the base-3 terms 1 2^{1+4*j} then 2 0^{2+4*j} for j in 0..oo, where ^ denotes repeated concatenation. This has been tested empirically up to j = 4300.

Michael S. Branicky, Table of n, a(n) for n = 1..10000
Eric Angelini, Michael S. Branicky, Giovanni Resta, N. J. A. Sloane, and David W. Wilson, The Comma Sequence: A Simple Sequence With Bizarre Properties, arXiv:2401.14346, Youtube
Michael S. Branicky, Bifurcation points on the road to infinity in the base-3 comma-child graph, starting at 1
Giovanni Resta, Graphical representation of a portion of the graph

Cf. A367355, A367618, A367619, A367620.

cp's OEIS Frontend

A367619 a(n) is the most remote positive ancestor of n in the comma-child graph in base 3.

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