cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A367353 Analog of A121805, but starting with 9.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, Nov 17 2023

Keywords

Comments

Contains 19511030 terms, the last being 999999936.

Links

Crossrefs

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.

Programs

  • Mathematica
    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 *)
  • Python
    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

Views

Author

N. J. A. Sloane, Nov 17 2023

Keywords

Comments

Contains 20573 terms, the last of which is 999936.

Links

Crossrefs

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.

Programs

  • Mathematica
    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 *)
  • Python
    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

A366487 First differences of "commas" sequence A121805.

Original entry on oeis.org

11, 23, 59, 41, 51, 62, 83, 13, 43, 74, 14, 55, 5, 55, 5, 56, 16, 77, 47, 18, 99, 89, 71, 81, 91, 1, 11, 21, 31, 41, 51, 61, 71, 81, 91, 1, 11, 21, 31, 41, 51, 61, 71, 82, 2, 22, 42, 62, 82, 2, 22, 42, 62, 82, 2, 22, 42, 62, 82, 2, 22, 42, 62, 82, 2, 22, 42, 63, 93, 23, 53, 83, 13, 43, 73, 3, 33, 63, 93, 23, 53, 83, 13, 43, 73, 3, 33, 63, 94, 34, 74, 14, 54, 94, 34, 74, 14, 54, 94
Offset: 1

Views

Author

N. J. A. Sloane, Nov 12 2023

Keywords

Comments

The record high points in this sequence are 11, 23, 59, 62, 83, 99, and they occur at terms 1, 2, 3, 6, 7, 21. Since 99 is the largest possible term, this is the full list of record high points.
The first differences of this sequence (i.e., the second differences of A121805) fall into the range [-90,90] for the first 99999 terms.
More generally, for a commas sequence in base b, the first differences are <= b^2 - 1. - Michael S. Branicky, Nov 16 2023.

Links

Crossrefs

Cf. A121805.
For the numbers missing from this sequence see A367349.

Programs

  • Python
    from itertools import islice
def agen(): # generator of terms
    an, y = 1, 1
    while y < 10:
        prevan = an
        an, y = an + 10*(an%10), 1
        while y < 10:
            if str(an+y)[0] == str(y):
                an += y
                break
            y += 1
        yield an - prevan
print(list(islice(agen(), 99))) # Michael S. Branicky, Nov 12 2023

A367344 Compute the commas sequence starting at 1, as in A121805, except do the calculations in octal. The terms are written here in octal (see also A367343).

Original entry on oeis.org

1, 12, 35, 106, 167, 261, 273, 326, 412, 436, 523, 560, 565, 643, 702, 731, 750, 757, 1050, 1051, 1062, 1103, 1134, 1175, 1246, 1327, 1420, 1421, 1432, 1453, 1504, 1545, 1616, 1677, 1770, 1771, 2003, 2035, 2107, 2201, 2213, 2245, 2317, 2411, 2423, 2455, 2527, 2621, 2633
Offset: 1

Views

Author

N. J. A. Sloane, Nov 15 2023

Keywords

Examples

			The first three terms are 1, 12, 35 in octal, and we check that around the first comma we see 1,1_8 which becomes 11_8 = 9_10, and 1_10 + 9_10 = 10_10 = 12_8 = a(2).
Around the second comma we see 2,3_8, which becomes 23_8 = 19_10, and indeed 12_8 + 23_8 = 10_10 + 19_10 = 29_10 = 35_8 = a(3).

Crossrefs

Cf. A121805, A367343.

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

Views

Author

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

Keywords

Comments

This is by definition a subsequence of A330128.

Links

Crossrefs

Cf. A037124, A121805, A330128, A330129, A367364, A367365.
For records see A367602 and A367603.

A367343 Compute the commas sequence starting at 1, as in A121805, except do the calculations in octal. The terms are written here in decimal (see also A367344).

Original entry on oeis.org

1, 10, 29, 70, 119, 177, 187, 214, 266, 286, 339, 368, 373, 419, 450, 473, 488, 495, 552, 553, 562, 579, 604, 637, 678, 727, 784, 785, 794, 811, 836, 869, 910, 959, 1016, 1017, 1027, 1053, 1095, 1153, 1163, 1189, 1231, 1289, 1299, 1325, 1367, 1425, 1435, 1461, 1503, 1562
Offset: 1

Views

Author

N. J. A. Sloane, Nov 15 2023

Keywords

Examples

			See A367344 for the calculation of the first three terms.

Links

Crossrefs

Cf. A121805, A367344.

Programs

  • Mathematica
    b = 8; a[1] = 1; a[n_] := a[n] = For[x = Mod[a[n - 1], b]; y = 0, y <= (b - 1), y++, k = a[n - 1] + b*x + y; If[y == IntegerDigits[k, b][[1]], Return[k]]]; Array[a, 10^4] (* Michael De Vlieger, Nov 15 2023, after Jean-François Alcover at A121805 *)
  • Python
    from itertools import islice
from sympy.ntheory.factor_ import digits
def agen(): # generator of terms
    an, y = 1, 1
    while y < 8:
        yield an
        an, y = an + 8*(an%8), 1
        while y < 8:
            if str(digits(an+y,8)[1]) == str(y):
                an += y
                break
            y += 1
print(list(islice(agen(), 52))) # Michael S. Branicky, Nov 16 2023

A365872 The "double commas" sequence, a variant of A121805. See the Comments and Example sections for detailed explanations.

Original entry on oeis.org

1, 25, 127, 271, 295, 403, 471, 499, 691, 725, 841, 877, 1019, 1201, 1223, 1285, 1387, 1529, 1711, 1733, 1795, 1897, 2041, 2065, 2169, 2353, 2417, 2561, 2585, 2689, 2873, 2937, 3083, 3149, 3335, 3441, 3467, 3613, 3679, 3865, 3971, 3997, 4145, 4253, 4321, 4349, 4537, 4685, 4793, 4861, 4889, 5079, 5269
Offset: 1

Views

Author

Eric Angelini and Giorgos Kalogeropoulos, Sep 21 2023

Keywords

Comments

The pair of digits adjacent to the comma between two terms forms an integer that is half the difference between the said terms. This is the lexicographically earliest sequence with this property. It will stop at some point, but when?

Examples

			a(1) = 1 and a(2) = 25 are separated by 24 units, and 24 is twice 12 (or 1,2);
a(2) = 25 and a(3) = 127 are separated by 102 units, and 102 is twice 51 (or 5,1);
a(3) = 127 and a(4) = 271 are separated by 144 units, and 144 is twice 72 (or 7,2);
a(4) = 271 and a(5) = 295 are separated by 24 units, and 24 is twice 12 (or 1,2); etc.

Links

Crossrefs

Cf. A121805 (the original 2006 sequence), A365873, A365874, A365875.

Programs

  • Mathematica
    a[1]=1;a[n_]:=a[n]=(k=a[n-1];While[2FromDigits@Join[{Mod[a[n-1],10]},{First@IntegerDigits@k}]!=k-a[n-1],k++];k);Array[a,70]

A365873 The "triple commas" sequence, a variant of A121805. See the Comments and Example sections for detailed explanations.

Original entry on oeis.org

1, 43, 136, 325, 487, 718, 985, 1138, 1381, 1414, 1537, 1750, 1753, 1846, 2032, 2098, 2344, 2470, 2476, 2662, 2728, 2974, 3103, 3202, 3271, 3310, 3319, 3598, 3847, 4069, 4351, 4393, 4495, 4657, 4879, 5164, 5299, 5584, 5719, 6007, 6235, 6403, 6511, 6559, 6847, 7078, 7339, 7630, 7651, 7702, 7783, 7894
Offset: 1

Views

Author

Eric Angelini and Giorgos Kalogeropoulos, Sep 21 2023

Keywords

Comments

The pair of digits adjacent to the comma between two terms forms an integer that is the third of the difference between the said terms. This is the lexicographically earliest sequence with this property. It will stop at some point, but when?

Examples

			a(1) = 1 and a(2) = 43 are separated by 42 units, and 42 is 3*14 (or 1,4);
a(2) = 43 and a(3) = 136 are separated by 93 units, and 93 is 3*31 (or 3,1);
a(3) = 136 and a(4) = 325 are separated by 189 units, and 189 is 3*63 (or 6,3);
a(4) = 325 and a(5) = 487 are separated by 162 units, and 162 is 3*54 (or 5,4); etc.

Links

Crossrefs

Cf. A121805 (the original 2006 sequence), A365872, A365874, A365875.

Programs

  • Mathematica
    a[1]=1;a[n_]:=a[n]=(k=a[n-1];While[3FromDigits@Join[{Mod[a[n-1],10]},{First@IntegerDigits@k}]!=k-a[n-1],k++];k);Array[a,70]

A365874 The "quadruple commas" sequence, a variant of A121805. See the Comments and Example sections for detailed explanations.

Original entry on oeis.org

1, 65, 273, 409, 797, 1081, 1125, 1329, 1693, 1817, 2105, 2313, 2441, 2489, 2857, 3149, 3521, 3573, 3705, 3917, 4213, 4349, 4725, 4941, 4997, 5297, 5597, 5897, 6201, 6265, 6489, 6873, 7021, 7089, 7477, 7785, 8017, 8329, 8721, 8793, 8945, 9181, 9257, 9573, 9729, 10093, 10217, 10501, 10545, 10749, 11113
Offset: 1

Views

Author

Eric Angelini and Giorgos Kalogeropoulos, Sep 21 2023

Keywords

Comments

The pair of digits adjacent to the comma between two terms forms an integer that is the fourth of the difference between the said terms. This is the lexicographically earliest sequence with this property. It will stop at some point, but when?

Examples

			a(1) = 1 and a(2) = 65 are separated by 64 units, and 64 is 4*16 (or 1,6);
a(2) = 65 and a(3) = 273 are separated by 208 units, and 208 is 4*52 (or 5,2);
a(3) = 273 and a(4) = 409 are separated by 136 units, and 136 is 4*34 (or 3,4);
a(4) = 409 and a(5) = 797 are separated by 388 units, and 388 is 4*97 (or 9,7); etc.

Links

Crossrefs

Cf. A121805 (the original 2006 sequence), A365872, A365873, A365875.

Programs

  • Mathematica
    a[1]=1;a[n_]:=a[n]=(k=a[n-1];While[4FromDigits@Join[{Mod[a[n-1],10]},{First@IntegerDigits@k}]!=k-a[n-1],k++];k);Array[a,70]

A365875 The "reversed commas" sequence, a variant of A121805. See the Comments and Example sections for detailed explanations.

Original entry on oeis.org

1, 92, 104, 118, 136, 152, 164, 178, 206, 232, 254, 278, 316, 352, 384, 428, 476, 532, 584, 648, 726, 812, 904, 998, 1016, 1032, 1044, 1058, 1076, 1092, 1104, 1118, 1136, 1152, 1164, 1178, 1196, 1212, 1224, 1238, 1256, 1272, 1284, 1298, 1316, 1332, 1344, 1358, 1376, 1392, 1404, 1418, 1436, 1452, 1464
Offset: 1

Views

Author

Eric Angelini and Giorgos Kalogeropoulos, Sep 21 2023

Keywords

Comments

The pair of digits adjacent to the comma between two terms forms an integer that is the difference between the said terms, but read backwards. This is the lexicographically earliest sequence with this property, provide the largest next term is always chosen when there is a choice. The sequence ends after 2514 steps with a(2514) = 99952.

Examples

			After a(1) = 1, we have multiple choices for a(2); they are 12, 22, 32, 42, 52, 62, 72, 82 and 92. We keep 92 for a(2) as 92 is the largest term.
a(1) = 1 and a(2) = 92 are separated by 91 units, and 91 is 19 backwards (or 1,9);
a(2) = 92 and a(3) = 104 are separated by 12 units, and 12 is 21 backwards (or 2,1);
a(3) = 104 and a(4) = 118 are separated by 14 units, and 14 is 41 backwards (or 4,1);
a(4) = 118 and a(5) = 136 are separated by 18 units, and 18 is 81 backwards (or 8,1); etc.
The next choices we encountered (keeping the largest term) were after a(8) = 178 -> {196,206}, then after 812 -> {894,904}, 1976 -> {1992,2002}, 3956 -> {3992,4002}, 19984 -> {19998,20008}, 39958 -> {39996,40006}, 49952 -> {49994,50004}, etc.

Links

Crossrefs

Cf. A121805 (the original 2006 sequence), A365872, A365873, A365874.

Programs

  • Mathematica
    NestList[(s=#;li=First@*IntegerDigits/@Table[s+10*k+Mod[s,10],{k,9}]-Range@9;
Last@Table[s+FromDigits@Join[Position[li,0][[c]],{Mod[s,10]}],{c,Length@Position[li,0]}])&,1,2513]
Previous Showing 11-20 of 72 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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