A367617 -id:A367617

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A367619

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

+10
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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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

FORMULA

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

PROG

(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)])

CROSSREFS

Cf. A367338, A367355, A367617, A367618.

KEYWORD

nonn,base

AUTHOR

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

STATUS

approved

A367366

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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.

LINKS

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

EXAMPLE

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.

PROG

(Python)

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