A367338 -id:A367338 - cp's OEIS frontend

A367616 a(n) is the unique k such that n is a comma-child of k, or -1 if k does not exist.

-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 10, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 20, 11, 2, -1, -1, -1, -1, -1, -1, -1, -1, 30, 21, 12, 3, -1, -1, -1, -1, -1, -1, -1, 40, 31, 22, 13, 4, -1, -1, -1, -1, -1, -1, 50, 41, 32, 23, 14, 14, 5, -1, -1, -1, -1, 60, 51, 42, 33, 33, 24, 15, 6, -1, -1, -1, 70, 61, 52, 52, 43, 34, 25, 16, 7

Offset: 1

N. J. A. Sloane, Dec 18 2023

Similar to A367614, but here we give the k such that n is a comma-child of k, whereas in A367614 n has to be a comma-successor of k. See A367338 for definitions.
The first difference between A367614 and the present sequence arises because 14 has one comma-successor, 59, but has two comma-children, 59 and 60. So A367614(59) = 14, A367614(60) = -1, while in the present sequence we have a(59) = a(60) = 14.
There are similar differences at n = 69 and 70, because both are comma-children of 33, and at many other places.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A121805, A367338, A367346, A367614.

def a(n):
    y = int(str(n)[0])
    x = (n-y)%10
    k = n - y - 10*x
    return k if k > 0 else -1
print([a(n) for n in range(1, 86)]) # Michael S. Branicky, Dec 18 2023

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

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, 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).

A367619 a(n) is the most remote positive ancestor of n in the comma-child graph in base 3.

Original entry on oeis.org

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

A367612 Numbers that are the comma-child of exactly one positive number.

Original entry on oeis.org

11, 12, 22, 23, 24, 33, 34, 35, 36, 44, 45, 46, 47, 48, 55, 56, 57, 58, 59, 60, 61, 66, 67, 68, 69, 70, 71, 72, 73, 77, 78, 79, 80, 81, 82, 83, 84, 85, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122

Offset: 1

Michael S. Branicky and N. J. A. Sloane, Dec 15 2023

This is the complement of A367611.
See A367338 for definition of comma-child.
May also be called numbers that have a positive comma-predecessor.

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, A367611.

def ok(n): y = int(str(n)[0]); x = (n-y)%10; return n - y - 10*x > 0
print([k for k in range(1, 123) if ok(k)]) # Michael S. Branicky, Dec 15 2023

A367620 The lexicographically earliest infinite sequence of positive numbers in which each term is a comma-child of the previous term.

Original entry on oeis.org

20, 22, 46, 107, 178, 260, 262, 284, 327, 401, 415, 469, 564, 610, 616, 682, 709, 807, 885, 944, 993, 1024, 1065, 1116, 1177, 1248, 1329, 1420, 1421, 1432, 1453, 1484, 1525, 1576, 1637, 1708, 1789, 1880, 1881, 1892, 1913, 1944, 1985, 2037, 2109, 2201, 2213, 2245, 2297, 2369, 2461, 2473, 2505, 2557, 2629, 2721, 2733, 2765

Offset: 1

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

Discovered by David W. Wilson in 2007 (see 2016 Angelini link).
The first choice point occurs for the term after a(412987860) = 19999999918, which has two comma-children.
We do not know which choice to take at that point. We do know by König's Infinity Lemma that one or both forks will extend to infinity. The definition of this sequence requires that we choose the smallest fork that has an infinite continuation.
Update, Dec 22 2023: We now know that the start of this sequence is one of four candidates (all other possible starts having terminated). The shortest of the four possible starts has length
   8278670191169895553395510925614764265575448369172463113087634743486440833078554
In other words, we know that there are only four possibilities for the initial prefix of that length.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000
Eric Angelini, The Commas Sequence, Message to Sequence Fans, Sep 06 2016. [Cached copy, with permission]
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, A367338, A367346.

A367622 Number of comma-children of n in base 10.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

N. J. A. Sloane, Dec 23 2023

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A121805, A367338 (definition), A367341 (0's), A367346 (2's).

f[n_]:=Module[{k=n+10*Last[IntegerDigits[n]]+Range[9]},Length[Select[k,#-n==
FromDigits[{Last[IntegerDigits[n]],First[IntegerDigits[#]]}]&]]]; f/@Range[108] (* Ivan N. Ianakiev, Dec 24 2023 *)

def a(n):
    x = 10*(n%10)
    return len([y for y in range(1, 10) if str(n+x+y)[0] == str(y)])
print([a(n) for n in range(1, 95)]) # Michael S. Branicky, Dec 23 2023

A367611 Numbers that are not the comma-child of any positive number.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 14, 15, 16, 17, 18, 19, 20, 21, 25, 26, 27, 28, 29, 30, 31, 32, 37, 38, 39, 40, 41, 42, 43, 49, 50, 51, 52, 53, 54, 62, 63, 64, 65, 74, 75, 76, 86, 87, 98

Offset: 1

Michael S. Branicky and N. J. A. Sloane, Dec 15 2023

A subsequence of A367600.
This 50-term sequence was found by David W. Wilson in 2007. See the Eric Angelini link.
See A367338 for definition of comma-child.

Eric Angelini, The Commas Sequence, Message to Sequence Fans, Sep 06 2016. [Cached copy, with permission]
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, A367600.

A367612 gives the complement.

def ok(n): y = int(str(n)[0]); x = (n-y)%10; return n - y - 10*x < 1
print([k for k in range(1, 99) if ok(k)]) # Michael S. Branicky, Dec 15 2023

A367613 Numbers with exactly one comma-child.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 53, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83

Offset: 1

N. J. A. Sloane, Dec 15 2023

Complement of union of A367341 and A367346.

See A367338 for definition of comma-child.

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. A121895, A367341 (numbers with no comma-children), A367346 (numbers with two comma-children).

fQ[n_]:=Module[{k=n+10*Last[IntegerDigits[n]]+Range[9]},Length[Select[k,#-n==FromDigits[{Last[IntegerDigits[n]],First[IntegerDigits[#]]}]&]]]==1;
Select[Range[83],fQ[#]&] (* Ivan N. Ianakiev, Dec 16 2023 *)

def ok(n):
    m = n + 10*(n%10)
    return len([m+y for y in range(1, 10) if int(str(m+y)[0]) == y]) == 1
print([k for k in range(1, 100) if ok(k)]) # Michael S. Branicky, Dec 28 2023

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A367616 a(n) is the unique k such that n is a comma-child of k, or -1 if k does not exist.

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

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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.

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A367619 a(n) is the most remote positive ancestor of n in the comma-child graph in base 3.

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

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A367612 Numbers that are the comma-child of exactly one positive number.

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A367620 The lexicographically earliest infinite sequence of positive numbers in which each term is a comma-child of the previous term.

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A367622 Number of comma-children of n in base 10.

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A367611 Numbers that are not the comma-child of any positive number.

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