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

%I #14 Feb 07 2024 13:21:43

%S 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,

%T 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,

%U 1,1,1,1,7,1,1,1,1,7,1,7,1,1,1,1,1,1,7

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

%C Analogous to A367617, but the calculations are done in base 3.

%C See A367338 for definitions of comma-child.

%C 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).

%H Michael S. Branicky, <a href="/A367619/b367619.txt">Table of n, a(n) for n = 1..2200</a>

%H Eric Angelini, Michael S. Branicky, Giovanni Resta, N. J. A. Sloane, and David W. Wilson, The Comma Sequence: A Simple Sequence With Bizarre Properties, <a href="http://arxiv.org/abs/2401.14346">arXiv:2401.14346</a>, <a href="https://www.youtube.com/watch?v=_EHAdf6izPI">Youtube</a>

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

%o (Python)

%o from functools import cache

%o from sympy.ntheory.factor_ import digits

%o def comma_parent(n, base=3): # A367618(n)

%o y = digits(n, base)[1]

%o x = (n-y)%base

%o k = n - y - base*x

%o return k if k > 0 else -1

%o @cache

%o def a(n):

%o cp = comma_parent(n)

%o if cp <= 0: return n

%o return a(cp)

%o print([a(n) for n in range(1, 88)])

%Y Cf. A367338, A367355, A367617, A367618.

%K nonn,base

%O 1,2

%A _Michael S. Branicky_ and _N. J. A. Sloane_, Dec 20 2023