A367355 -id:A367355 - cp's OEIS frontend

A367356 Length of base-3 Commas sequence when started at n.

17, 5, 2, 1, 16, 164, 490, 163, 4, 3, 489, 15, 14, 2, 162, 161, 13, 1472, 488, 1471, 160, 1, 159, 487, 12, 486, 1470, 1469, 11, 158, 157, 1468, 485, 484, 156, 10, 9, 483, 1467, 1466, 8, 155, 154, 1465, 482, 481, 153, 7, 6, 480, 1464, 1463, 5, 41, 152, 40, 1462, 479, 1461, 151, 4, 150, 478, 39, 477, 3, 1460, 2, 38, 149, 37, 1459, 476, 1458, 148, 1, 147, 475, 36, 474, 1457, 1456, 35, 146, 145, 1455, 473, 472, 144, 34, 33, 471, 1454, 1453, 32, 143, 142, 1452, 470, 469

Offset: 1

N. J. A. Sloane, Nov 18 2023

a(n) = 1 for n = 4, 22, 76, ... (the numbers 222...2211 in ternary)

We now know that a(n) is finite for all n.

			For a(1) = 17, see A367355, which has 17 terms.

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, A330128, A367355.

from itertools import islice
from sympy.ntheory.factor_ import digits
def a(n, b=3): # generator of terms
    an, y, c = n, 1, 0
    while y < b:
        an, y, c = an + b*(an%b), 1, c+1
        while y < b:
            if str(digits(an+y, b)[1]) == str(y):
                an += y
                break
            y += 1
    return c
print([a(n) for n in range(1, 101)]) # Michael S. Branicky, Nov 18 2023

A367606 Comma-successor to n working in base 3, but written in base 10, or -1 if n has no successor.

Original entry on oeis.org

5, 9, 4, -1, 12, 8, 11, 15, 10, 14, 19, 13, 17, 22, 16, 21, 25, 20, 24, 27, 23, -1, 30, 26, 29, 33, 28, 32, 36, 31, 35, 39, 34, 38, 42, 37, 41, 45, 40, 44, 48, 43, 47, 51, 46, 50, 55, 49, 53, 58, 52, 57, 61, 56, 60, 64, 59, 63, 67, 62, 66, 70, 65, 69, 73, 68, 72, 76, 71, 75, 79, 74, 78, 81, 77, -1, 84, 80, 83, 87, 82

Offset: 1

N. J. A. Sloane, Dec 11 2023

This is a base-3 analog of A367338.
It seems that the indices of the terms equal to -1 are in A168613. - Ivan N. Ianakiev, Dec 12 2023
This is true for A168613(n), n >= 2.  See proofs in A367341. - Michael S. Branicky, Dec 15 2023

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

Cf. A121085, A168613, A367338, A367341, A367355, A367356, A367607, A367608, A367609.

from sympy.ntheory.factor_ import digits
def a(n):
    b = n + 3*(n%3)
    return next((b+y for y in [1, 2] if digits(b+y, 3)[1] == y), -1)
print([a(n) for n in range(1, 82)]) # Michael S. Branicky, Dec 11 2023

A367609 Comma-number associated with A367607(n), and written in base 3, or -1 if A367607(n) = -1.

Original entry on oeis.org

11, 21, 1, -1, 21, 2, 11, 21, 1, 11, 22, 1, 11, 22, 1, 12, 22, 2, 12, 21, 2, -1, 21, 2, 11, 21, 1, 11, 21, 1, 11, 21, 1, 11, 21, 1, 11, 21, 1, 11, 21, 1, 11, 21, 1, 11, 22, 1, 11, 22, 1, 12, 22, 2, 12, 22, 2, 12, 22, 2, 12, 22, 2, 12, 22, 2, 12, 22, 2, 12, 22, 2, 12, 21, 2, -1, 21, 2, 11, 21, 1, 11, 21, 1, 11, 21, 1, 11, 21, 1, 11, 21, 1, 11, 21, 1, 11, 21, 1, 11

Offset: 1

N. J. A. Sloane, Dec 11 2023

This is a base-3 analog of A367339.

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. A121085, A367339, A367355, A367356, A367606-A367608.

from sympy.ntheory.factor_ import digits
def a(n):
    b = n + 3*(n%3)
    return next((int("".join(map(str, digits(b+y-n, 3)[1:]))) for y in [1, 2] if digits(b+y, 3)[1] == y), -1)
print([a(n) for n in range(1, 101)]) # Michael S. Branicky, Dec 11 2023

A367608 Comma-number associated with A367606(n), but written in base 10, or -1 if A367606(n) = -1.

Original entry on oeis.org

4, 7, 1, -1, 7, 2, 4, 7, 1, 4, 8, 1, 4, 8, 1, 5, 8, 2, 5, 7, 2, -1, 7, 2, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 8, 1, 4, 8, 1, 5, 8, 2, 5, 8, 2, 5, 8, 2, 5, 8, 2, 5, 8, 2, 5, 8, 2, 5, 8, 2, 5, 7, 2, -1, 7, 2, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4

Offset: 1

N. J. A. Sloane, Dec 11 2023

If n has a comma-successor m (say) in base 3, then a(n) is the comma-number linking n and m, and is equal to m-n; a(n) = -1 if n has no successor. See A367338 for definitions.

This is a base-3 analog of A367339.

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

Cf. A121085, A367339, A367355, A367356, A367606-A367609.

from sympy.ntheory.factor_ import digits
def a(n):
    b = n + 3*(n%3)
    return next((b+y-n for y in [1, 2] if digits(b+y, 3)[1] == y), -1)
print([a(n) for n in range(1, 101)]) # Michael S. Branicky, Dec 11 2023

A367607 Comma-successor to n working in base 3, and written in base 3, or -1 if n has no successor.

Original entry on oeis.org

12, 100, 11, -1, 110, 22, 102, 120, 101, 112, 201, 111, 122, 211, 121, 210, 221, 202, 220, 1000, 212, -1, 1010, 222, 1002, 1020, 1001, 1012, 1100, 1011, 1022, 1110, 1021, 1102, 1120, 1101, 1112, 1200, 1111, 1122, 1210, 1121, 1202, 1220, 1201, 1212, 2001, 1211, 1222, 2011, 1221, 2010, 2021, 2002, 2020, 2101, 2012, 2100

Offset: 1

N. J. A. Sloane, Dec 11 2023

This is a base-3 analog of A367338.

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

Cf. A121085, A007089, A367338, A367355, A367356, A367606, A367608, A367609.

from sympy.ntheory.factor_ import digits
def a(n):
    b = n + 3*(n%3)
    return next((int("".join(map(str, d3))) for y in [1, 2] if (d3:=digits(b+y, 3)[1:])[0] == y), -1)
print([a(n) for n in range(1, 61)]) # Michael S. Branicky, Dec 11 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)).

A367598 Record high-points in A367356.

Original entry on oeis.org

17, 164, 490, 1472, 39819, 119228, 3225387, 9657464, 261256395, 782254580, 21161768043

Offset: 1

N. J. A. Sloane, Nov 24 2023

This is the base-3 analog of A367364.

Cf. A121805, A367355, A367356, A367364, A367599.

a(7)-a(11) from Michael S. Branicky, Nov 27 2023

A367599 Indices of record high-points in A367356.

Original entry on oeis.org

1, 6, 7, 18, 162, 1458, 13122, 118098, 1062882, 9565938, 86093442

Offset: 1

N. J. A. Sloane, Nov 24 2023

This is a base-3 analog of A367365. The present sequence includes the terms 2*3, 2*3^2, 2*3^6, whereas A367365 includes 4*10 and 4*10^3.

Terms a(4)-a(11) are of the form 2*3^(2*i), i > 0. - Michael S. Branicky, Nov 26 2023

Cf. A121805, A367355, A367356, A367364, A367598.

a(7)-a(11) from Michael S. Branicky, Nov 27 2023

A367605 Final term of commas sequence (cf. A121805) if start at 1 and do the calculations in base n; or -1 if the sequence is infinite.

Original entry on oeis.org

-1, 76, 6, 15612, 60466165, 823512, 262122, 32, 99999945, 1771460, 110, 2052, 289254654871, 8649755859206, 18446744073709551480, 83264, 1338258845052394702439737982907, 893871504, 10239999999942, 1801088480, 189, 148035426, 13501, 244140456, 3670344486987375

Offset: 2

N. J. A. Sloane, Dec 08 2023

sign

a(n) is written here in base 10. In base n the values are more revealing: they are -1, 2211_3, 12_4, 444422_5, 5555555541_6, 6666624_7, 777752_8, 35_9, and 99999945_10. That is, they consist of a possibly empty string of digits b-1 followed by a pair of digits xy with x+y = b-1 (see the theorem in A367341).

Cf. A121805, A367341, A367604.

The sequences for bases 3, 8, and 10 are A367355, A367344, and A121805.

More terms from Michael S. Branicky, Dec 08 2023

cp's OEIS Frontend

A367356 Length of base-3 Commas sequence when started at n.

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A367606 Comma-successor to n working in base 3, but written in base 10, or -1 if n has no successor.

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A367609 Comma-number associated with A367607(n), and written in base 3, or -1 if A367607(n) = -1.

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A367608 Comma-number associated with A367606(n), but written in base 10, or -1 if A367606(n) = -1.

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A367607 Comma-successor to n working in base 3, and written in base 3, or -1 if n has no successor.

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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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A367598 Record high-points in A367356.

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A367599 Indices of record high-points in A367356.

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