A330128 -id:A330128 - cp's OEIS frontend

A367353 Analog of A121805, but starting with 9.

9, 100, 101, 112, 133, 164, 206, 268, 351, 364, 408, 492, 517, 592, 618, 705, 762, 789, 887, 966, 1027, 1098, 1179, 1270, 1271, 1282, 1303, 1334, 1375, 1426, 1487, 1558, 1639, 1730, 1731, 1742, 1763, 1794, 1835, 1886, 1947, 2019, 2111, 2123, 2155, 2207, 2279, 2371

Offset: 1

N. J. A. Sloane, Nov 17 2023

Contains 19511030 terms, the last being 999999936.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Comma sequences in base 10, starting with 1, 2, 4, 5, 6, 7, 8, 9, 10 are A121805, A139284, A366492, A367337, A367350, A367351, A367352, A367353, A367354. Starting with 3 is trivial, and those starting with 11, 12, 13 are essentially duplicates.

Cf. A330128, A330129.

b = 10; m = b - 1; a[1] = 9; a[n_] := a[n] = For[r = Mod[a[n - 1], b]; y = 0, y <= m, y++, If[y == IntegerDigits[#, b][[1]], Return[#]] &[a[n - 1] + b r + y]]; Array[a, 45] (* Michael De Vlieger, Nov 18 2023, after Jean-François Alcover at A121805 *)

from itertools import islice
def agen(start=9): # generator of terms
    an, y = start, 1
    while y < 10:
        yield an
        an, y = an + 10*(an%10), 1
        while y < 10:
            if str(an+y)[0] == str(y):
                an += y
                break
            y += 1
print(list(islice(agen(), 50))) # Michael S. Branicky, Nov 18 2023

A367354 Analog of A121805, but starting with 10.

Original entry on oeis.org

10, 11, 23, 58, 139, 231, 243, 275, 328, 412, 436, 501, 516, 581, 596, 662, 688, 775, 833, 871, 889, 988, 1069, 1160, 1161, 1172, 1193, 1224, 1265, 1316, 1377, 1448, 1529, 1620, 1621, 1632, 1653, 1684, 1725, 1776, 1837, 1908, 1989, 2081, 2093, 2125, 2177, 2249, 2341, 2353

Offset: 1

N. J. A. Sloane, Nov 17 2023

Contains 20573 terms, the last of which is 999936.

Michael De Vlieger, Table of n, a(n) for n = 1..20573

Comma sequences in base 10, starting with 1, 2, 4, 5, 6, 7, 8, 9, 10 are A121805, A139284, A366492, A367337, A367350, A367351, A367352, A367353, A367354. Starting with 3 is trivial, and those starting with 11, 12, 13 are essentially duplicates.

Cf. A330128, A330129.

a[1] = b = 10; m = b - 1; a[n_] := a[n] = For[r = Mod[a[n - 1], b]; y = 0, y <= m, y++, If[y == IntegerDigits[#, b][[1]], Return[#]] &[a[n - 1] + b r + y]]; Array[a, 50] (* Michael De Vlieger, Nov 18 2023, after Jean-François Alcover at A121805 *)

from itertools import islice
def agen(start=10): # generator of terms
    an, y = start, 1
    while y < 10:
        yield an
        an, y = an + 10*(an%10), 1
        while y < 10:
            if str(an+y)[0] == str(y):
                an += y
                break
            y += 1
print(list(islice(agen(), 50))) # Michael S. Branicky, Nov 18 2023

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

Original entry on oeis.org

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

A367601 a(n) is the number of terms in the analog of A121805 but starting with A037124(n), or -1 if that sequence is infinite.

Original entry on oeis.org

2137453, 194697747222394, 2, 199900, 19706, 209534289952018960, 15, 198104936410, 19511030, 20573, 19278442756937613, 207556412347088426, 2153441655319779164332, 1960210914, 204, 19, 195607586, 21, 19511029, 1922379655900, 15, 1979, 191782579276710577865, 1927

Offset: 1

Michael S. Branicky and N. J. A. Sloane, Dec 06 2023

This is by definition a subsequence of A330128.

Michael S. Branicky, Table of n, a(n) for n = 1..1000
Michael S. Branicky, Table of indices and record highs in A367601

Cf. A037124, A121805, A330128, A330129, A367364, A367365.

For records see A367602 and A367603.

A367602 Records in A367601.

Original entry on oeis.org

2137453, 194697747222394, 209534289952018960, 2153441655319779164332, 195152998207833388640389, 192648330068920004741771823742285752, 1879472501974027932230497653831908067612145407102, 2071675282852490774827341955075117685752805692835677843166, 20548999112584138590755517725134777010151822745525893951682

Offset: 1

Michael S. Branicky, Dec 06 2023

Is this the same as A367364? - R. J. Mathar, Dec 12 2023

Michael S. Branicky, Table of n, a(n) for n = 1..133
Michael S. Branicky, Table of indices and record highs in A367601

Cf. A121805, A330128, A330129, A367364, A367365, A367603.

A367603 Indices of records in A367601.

Original entry on oeis.org

1, 2, 6, 40, 4000, 20000, 80000000000, 40000000000000000000000000000, 30000000000000000000000000000000, 3000000000000000000000000000000000000000, 6000000000000000000000000000000000000000

Offset: 1

Michael S. Branicky, Dec 06 2023

Michael S. Branicky, Table of n, a(n) for n = 1..133
Michael S. Branicky, Table of indices and record highs in A367601

Cf. A121805, A330128, A330129, A367364, A367365, A367602.

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

A367353 Analog of A121805, but starting with 9.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

A367354 Analog of A121805, but starting with 10.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

A367601 a(n) is the number of terms in the analog of A121805 but starting with A037124(n), or -1 if that sequence is infinite.

Views

Author

Keywords

Comments

Links

Crossrefs

A367602 Records in A367601.

Views

Author

Keywords

Comments

Links

Crossrefs

A367603 Indices of records in A367601.

Views

Author

Keywords

Links

Crossrefs

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python