A367616 -id:A367616 - cp's OEIS frontend

A367617 a(n) is the most remote positive ancestor of n in the comma-child graph.

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, 14, 5, 62, 63, 64, 65, 14, 51, 42, 30, 30, 2, 15, 6, 74, 75

Offset: 1

Michael S. Branicky and N. J. A. Sloane, Dec 18 2023

Like A367366 but allows ancestors that are not comma-predecessors. More specifically, A367366(n) is the most remote positive ancestor of n in the comma-successor graph. See A367338 for definitions.

This sequence first differs from A367366 at n = 60.

			a(60) = a(66) = 14, since 66 is a comma-child of 60, and 60 is a comma-child of 14, and 14 is not the comma-child of any positive number. In other words, A367616(A367616(66)) = A367616(60) = 14, and A367616(14) = -1.

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

Cf. A367338, A367366, A367616.

def comma_parent(n): # A367616(n)
    y = int(str(n)[0])
    x = (n-y)%10
    k = n - y - 10*x
    return k if k > 0 else -1
def a(n):
    an = n
    while (cp:=comma_parent(an)) > 0: an = cp
    return an
print([a(n) for n in range(1, 76)]) # Michael S. Branicky, Dec 18 2023

A367618 a(n) is the unique k such that n is a comma-child of k in base 3, or -1 if k does not exist.

Original entry on oeis.org

-1, -1, -1, 3, 1, 1, -1, 6, 2, 9, 7, 5, 12, 10, 8, 15, 13, 13, 11, 18, 16, 14, 21, 19, 17, 24, 20, 27, 25, 23, 30, 28, 26, 33, 31, 29, 36, 34, 32, 39, 37, 35, 42, 40, 38, 45, 43, 41, 48, 46, 44, 51, 49, 49, 47, 54, 52, 50, 57, 55, 53, 60, 58, 56, 63, 61, 59, 66, 64, 62, 69, 67, 65, 72, 70, 68, 75, 73, 71, 78, 74, 81, 79, 77, 84, 82

Offset: 1

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

Analogous to A367616, but the calculations are done in base 3.
See A367338 for definitions of comma-child.
May also be called the "comma-parent" of n since n is the comma-child of a(n).

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

Cf. A367338, A367355, A367616, A367619.

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

a(n) = n - y - b*((n-y) mod b) where b is the base and y is the first digit of a(n); it is said to exist if a(n) > 0, else undefined (here, -1).

cp's OEIS Frontend

A367617 a(n) is the most remote positive ancestor of n in the comma-child graph.

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