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

Author

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

Keywords

• nonn
• base

Comments

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).

Links

• 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

Crossrefs

Cf. A367338, A367355, A367617, A367618.

Programs

Python

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)])

Formula

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