A367617 -id:A367617 - 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)).

A367366 a(n) = smallest k such that the commas sequence (cf. A121805) with initial term k contains n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 10, 1, 13, 14, 15, 16, 17, 18, 19, 20, 21, 20, 10, 2, 25, 26, 27, 28, 29, 30, 31, 32, 30, 21, 1, 3, 37, 38, 39, 40, 41, 42, 43, 40, 31, 20, 13, 4, 49, 50, 51, 52, 53, 54, 50, 41, 32, 10, 14, 60, 5, 62, 63, 64, 65, 60, 51, 42, 30, 70, 2, 15, 6, 74, 75

Offset: 1

N. J. A. Sloane, Dec 05 2023

Every k >= 1 appears in this sequence exactly A330128(k) times. So there are 2137453 1's, 194697747222394 2's, 2 3's, 209534289952018960 6's, and so on.

a(n) is the most remote ancestor of n in the comma-successor graph.

			All terms n in A121805 have a(n) = 1, all n in A139284 have a(n) = 2, all n in A366492 have a(n) = 4, and so on.

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, Fibonacci Quarterly 62:3 (2024), 215-232.
Eric Angelini, Michael S. Branicky, Giovanni Resta, N. J. A. Sloane, and David W. Wilson, The Comma Sequence: A Simple Sequence With Bizarre Properties, Local copy.
N. J. A. Sloane, Eric Angelini's Comma Sequence, Experimental Math Seminar, Rutgers Univ., January 18, 2024, Youtube video; Slides

Cf. A121805, A139284, A330128, A366492, A367338, A367341, A367617.

def comma_predecessor(n): # A367614(n)
    y = int(str(n)[0])
    x = (n-y)%10
    k = n - y - 10*x
    kk = k + 10*x + y-1
    return k if k > 0 and int(str(kk)[0]) != y-1 else -1
def a(n):
    an = n
    while (cp:=comma_predecessor(an)) > 0: an = cp
    return an
print([a(n) for n in range(1, 76)]) # Michael S. Branicky, Dec 18 2023

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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