A121041 -id:A121041 - cp's OEIS frontend

A355632 Irregular triangle T(n, k), n > 0, k = 1..A121041(n), read by rows; the n-th row contains in ascending order the divisors of n whose decimal expansions appear as substrings in the decimal expansion of n.

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 10, 1, 11, 1, 2, 12, 1, 13, 1, 14, 1, 5, 15, 1, 16, 1, 17, 1, 18, 1, 19, 2, 20, 1, 21, 2, 22, 23, 2, 4, 24, 5, 25, 2, 26, 27, 2, 28, 29, 3, 30, 1, 31, 2, 32, 3, 33, 34, 5, 35, 3, 6, 36, 37, 38, 3, 39, 4, 40, 1, 41, 2, 42, 43, 4, 44

Offset: 1

Rémy Sigrist, Jul 11 2022

			Triangle T(n, k) begins:
     1: [1]
     2: [2]
     3: [3]
     4: [4]
     5: [5]
     6: [6]
     7: [7]
     8: [8]
     9: [9]
    10: [1, 10]
    11: [1, 11]
    12: [1, 2, 12]
    13: [1, 13]
    14: [1, 14]
    15: [1, 5, 15]
    16: [1, 16]

Paolo Xausa, Table of n, a(n) for n = 1..12288 (rows 1..5000 of the triangle, flattened).
Index entries for sequences related to decimal expansion of n
Index entries for sequences related to divisors

Cf. A027750, A121041 (row lengths), A121042, A355620 (row sums), A355634 (binary analog).

Table[Select[Divisors[n], StringContainsQ[IntegerString[n], IntegerString[#]] &], {n, 50}] (* Paolo Xausa, Jul 23 2024 *)

row(n, base=10) = { my (d=digits(n, base), s=setbinop((i,j) -> fromdigits(d[i..j], base), [1..#d]), v=0); select(v -> v && n%v==0, s) }

from sympy import divisors
def row(n):
    s = str(n)
    return sorted(d for d in divisors(n, generator=True) if str(d) in s)
def table(r): return [i for n in range(1, r+1) for i in row(n)]
print(table(44)) # Michael S. Branicky, Jul 11 2022

T(n, 1) = A121042(n).
T(n, A121041(n)) = n.
Sum_{k = 1..A121041(n)} T(n, k) = A355620(n).

A011531 Numbers that contain a digit 1 in their decimal representation.

Original entry on oeis.org

1, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 31, 41, 51, 61, 71, 81, 91, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133

Offset: 1

N. J. A. Sloane

A121042(a(n)) = 1. - Reinhard Zumkeller, Jul 21 2006

See A043493 for numbers that contain a single digit '1'. A subsequence of numbers having a digit that divides all other digits, A263314. - M. F. Hasler, Jan 11 2016

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for 10-automatic sequences.

Cf. A121041, A121022, A121023, A121024, A121025, A121026, A121027, A121028, A121029, A121030, A121031, A121032, A121033, A121034, A121035, A121036, A121037, A121038, A121039, A121040.

Cf. A052383 (complement), A062634 (a subsequence).

Filtered([1..140],n->1 in ListOfDigits(n)); # Muniru A Asiru, Feb 23 2019

a011531 n = a011531_list !! (n-1)
a011531_list = filter ((elem '1') . show) [0..]
-- Reinhard Zumkeller, Feb 05 2012

[n: n in [0..500] | 1 in Intseq(n) ]; // Vincenzo Librandi, Jan 11 2016

M:= 3: # to get all terms of up to M digits
B:= {1}: A:= {1}:
for i from 2 to M do
   B:= map(t -> seq(10*t+j,j=0..9),B) union
      {seq(10*x+1,x=2*10^(i-2)..10^(i-1)-1)}:
   A:= A union B;
od:
sort(convert(A,list)); # Robert Israel, Jan 10 2016
# second program:
A011531 := proc(n)
    if n = 1 then
        1;
    else
        for a from procname(n-1)+1 do
            if nops(convert(convert(a,base,10),set) intersect {1}) > 0 then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Jul 31 2016

Select[Range[600] - 1, DigitCount[#, 10, 1] > 0 &] (* Vincenzo Librandi, Jan 11 2016 *)

is_A011531(n)=setsearch(Set(digits(n)),1) \\ M. F. Hasler, Jan 10 2016

def aupto(nn): return [m for m in range(1, nn+1) if '1' in str(m)]
print(aupto(133)) # Michael S. Branicky, Jan 10 2021

(0 to 119).filter(.toString.indexOf('1') > -1) // _Alonso del Arte, Jan 12 2020

a(n) ~ n. - Charles R Greathouse IV, Nov 02 2022

A121022 Even numbers containing a 2 in their decimal representation.

Original entry on oeis.org

2, 12, 20, 22, 24, 26, 28, 32, 42, 52, 62, 72, 82, 92, 102, 112, 120, 122, 124, 126, 128, 132, 142, 152, 162, 172, 182, 192, 200, 202, 204, 206, 208, 210, 212, 214, 216, 218, 220, 222, 224, 226, 228, 230, 232, 234, 236, 238, 240, 242, 244, 246, 248, 250, 252

Offset: 1

Reinhard Zumkeller, Jul 21 2006

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Seong Ju Kim, R. Stees, L. Taalman, Sequences of Spiral Knot Determinants, Journal of Integer Sequences, Vol. 19 (2016), # 16.1.4.
Ryan Stees, Sequences of Spiral Knot Determinants, Senior Honors Projects, Paper 84, James Madison Univ., May 2016.
Index entries for 10-automatic sequences.

Intersection of A005843 and A011532.

Cf. A121041, A011531, A121023, A121024, A121025, A121026, A121027, A121028, A121029, A121030, A121031, A121032, A121033, A121034, A121035, A121036, A121037, A121038, A121039, A121040.

a121022 n = a121022_list !! (n-1)
a121022_list = filter (('2' `elem`) . show) [0, 2 ..]
-- Reinhard Zumkeller, Nov 08 2013

Select[2*Range[200],DigitCount[#,10,2]>0&] (* Harvey P. Dale, Nov 04 2013 *)

is(n) = n%2==0 && setsearch(vecsort(digits(n)), 2) \\ Felix Fröhlich, Nov 22 2020

a(n) ~ 2n. - Charles R Greathouse IV, Mar 30 2022

A121040 Multiples of 20 containing a 20 in their decimal representation.

Original entry on oeis.org

20, 120, 200, 220, 320, 420, 520, 620, 720, 820, 920, 1020, 1120, 1200, 1220, 1320, 1420, 1520, 1620, 1720, 1820, 1920, 2000, 2020, 2040, 2060, 2080, 2120, 2200, 2220, 2320, 2420, 2520, 2620, 2720, 2820, 2920, 3020, 3120, 3200, 3220, 3320, 3420, 3520

Offset: 1

Reinhard Zumkeller, Jul 21 2006

Not the same as A044352.

Index entries for 10-automatic sequences.

Cf. A121041, A008602, A011531, A121022, A121023, A121024, A121025, A121026, A121027, A121028, A121029, A121030, A121031, A121032, A121033, A121034, A121035, A121036, A121037, A121038, A121039.

Select[20*Range[200],SequenceCount[IntegerDigits[#],{2,0}]>0&] (* The program uses the SequenceCount function from Mathematica version 10 *) (* Harvey P. Dale, Nov 27 2015 *)

is(n)=if(n%20, return(0)); while(n>19, if(n%100==20, return(1)); n\=10); 0 \\ Charles R Greathouse IV, Feb 12 2017

a(n) ~ 20n. - Charles R Greathouse IV, Feb 12 2017

Corrected by T. D. Noe, Oct 25 2006

A121025 Multiples of 5 containing a 5 in their decimal representation.

Original entry on oeis.org

5, 15, 25, 35, 45, 50, 55, 65, 75, 85, 95, 105, 115, 125, 135, 145, 150, 155, 165, 175, 185, 195, 205, 215, 225, 235, 245, 250, 255, 265, 275, 285, 295, 305, 315, 325, 335, 345, 350, 355, 365, 375, 385, 395, 405, 415, 425, 435, 445, 450, 455, 465, 475, 485

Offset: 1

Reinhard Zumkeller, Jul 21 2006

Paolo Xausa, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Muniru A Asiru)
Index entries for 10-automatic sequences.

Intersection of A008587 and A011535.

Cf. A121041, A011531, A121022, A121023, A121024, A121026, A121027, A121028, A121029, A121030, A121031, A121032, A121033, A121034, A121035, A121036, A121037, A121038, A121039, A121040.

Filtered([1..500],n-> n mod 5 = 0 and 5 in ListOfDigits(n)); # Muniru A Asiru, Feb 23 2019

Select[5*Range[200], MemberQ[IntegerDigits[#], 5] &] (* Paolo Xausa, Feb 25 2024 *)

is(n)=n%5==0 && setsearch(Set(digits(n)), 5) \\ Charles R Greathouse IV, Feb 12 2017

a(n) ~ 5n. - Charles R Greathouse IV, Feb 12 2017

Typo in comment fixed by Reinhard Zumkeller, May 01 2011

A121030 Multiples of 10 containing a 10 in their decimal representation.

Original entry on oeis.org

10, 100, 110, 210, 310, 410, 510, 610, 710, 810, 910, 1000, 1010, 1020, 1030, 1040, 1050, 1060, 1070, 1080, 1090, 1100, 1110, 1210, 1310, 1410, 1510, 1610, 1710, 1810, 1910, 2010, 2100, 2110, 2210, 2310, 2410, 2510, 2610, 2710, 2810, 2910, 3010, 3100

Offset: 1

Reinhard Zumkeller, Jul 21 2006

R. Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for 10-automatic sequences.

Cf. A121041, A008592, A011531, A121022, A121023, A121024, A121025, A121026, A121027, A121028, A121029, A121031, A121032, A121033, A121034, A121035, A121036, A121037, A121038, A121039, A121040.

Select[10*Range[1000], StringContainsQ[IntegerString[#], "10"] &] (* Paolo Xausa, Feb 25 2024 *)

a(n) ~ 10n. - Charles R Greathouse IV, Feb 12 2017

A121032 Multiples of 12 containing a 12 in their decimal representation.

Original entry on oeis.org

12, 120, 312, 612, 912, 1128, 1200, 1212, 1224, 1236, 1248, 1260, 1272, 1284, 1296, 1512, 1812, 2112, 2124, 2412, 2712, 3012, 3120, 3312, 3612, 3912, 4128, 4212, 4512, 4812, 5112, 5124, 5412, 5712, 6012, 6120, 6312, 6612, 6912, 7128, 7212, 7512, 7812

Offset: 1

Reinhard Zumkeller, Jul 21 2006

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for 10-automatic sequences.

Subsequence of A008594.

Cf. A121041, A011531, A121022, A121023, A121024, A121025, A121026, A121027, A121028, A121029, A121030, A121031, A121033, A121034, A121035, A121036, A121037, A121038, A121039, A121040.

import Data.List (isInfixOf)
a121032 n = a121032_list !! (n-1)
a121032_list = filter ((isInfixOf "12") . show) a008594_list
-- Reinhard Zumkeller, Dec 12 2012

Select[12*Range[700],SequenceCount[IntegerDigits[#],{1,2}]>0&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 25 2017 *)

is(n)=if(n%12, return(0)); while(n>11, if(n%100==12, return(1)); n\=10); 0 \\ Charles R Greathouse IV, Feb 12 2017

a(n) ~ 12n. - Charles R Greathouse IV, Nov 02 2022

Data fixed by Reinhard Zumkeller, Dec 12 2012

A121023 Multiples of 3 containing a 3 in their decimal representation.

Original entry on oeis.org

3, 30, 33, 36, 39, 63, 93, 123, 132, 135, 138, 153, 183, 213, 231, 234, 237, 243, 273, 300, 303, 306, 309, 312, 315, 318, 321, 324, 327, 330, 333, 336, 339, 342, 345, 348, 351, 354, 357, 360, 363, 366, 369, 372, 375, 378, 381, 384, 387, 390, 393, 396, 399

Offset: 1

Reinhard Zumkeller, Jul 21 2006

Intersection of A008585 and A011533.

The graph of this sequence is (roughly) self-similar: it has the same appearance when the scale is multiplied by 10. - M. F. Hasler, Mar 09 2014

M. F. Hasler, Table of n, a(n) for n = 1..3440
Index entries for 10-automatic sequences.

Cf. A121041, A011531, A121022, A121024, A121025, A121026, A121027, A121028, A121029, A121030, A121031, A121032, A121033, A121034, A121035, A121036, A121037, A121038, A121039, A121040.

Select[3*Range[500], MemberQ[IntegerDigits[#],3] &] (* Paolo Xausa, Feb 25 2024 *)

is(n)=!(n%3)&&setsearch(Set(digits(n)),3) \\ M. F. Hasler, Mar 09 2014

c=0;forstep(n=3,3e4,3,is(n)&write("/tmp/b121023.txt",c++" "n))

a(n) ~ 3n. - Charles R Greathouse IV, Mar 31 2016

Typo in comment fixed by Reinhard Zumkeller, May 01 2011

A121024 Multiples of 4 containing a 4 in their decimal representation.

Original entry on oeis.org

4, 24, 40, 44, 48, 64, 84, 104, 124, 140, 144, 148, 164, 184, 204, 224, 240, 244, 248, 264, 284, 304, 324, 340, 344, 348, 364, 384, 400, 404, 408, 412, 416, 420, 424, 428, 432, 436, 440, 444, 448, 452, 456, 460, 464, 468, 472, 476, 480, 484, 488, 492, 496

Offset: 1

Reinhard Zumkeller, Jul 21 2006

Intersection of A008586 and A011534.

Index entries for 10-automatic sequences.

Cf. A121041, A011531, A121022, A121023, A121025, A121026, A121027, A121028, A121029, A121030, A121031, A121032, A121033, A121034, A121035, A121036, A121037, A121038, A121039, A121040.

Select[4Range[150],DigitCount[#,10,4]>0&] (* Harvey P. Dale, Jun 11 2011 *)

is(n)=if(n%4,return(0));n=eval(Vec(Str(n)));for(i=1,#n,if(n[i]==4,return(1)));0 \\ Charles R Greathouse IV, Jul 16 2011

a(n) ~ 4n. - Charles R Greathouse IV, Jul 16 2011

Typo in comment fixed by Reinhard Zumkeller, May 01 2011

A121027 Multiples of 7 containing a 7 in their decimal representation.

Original entry on oeis.org

7, 70, 77, 147, 175, 217, 273, 287, 357, 371, 378, 427, 476, 497, 567, 574, 637, 672, 679, 700, 707, 714, 721, 728, 735, 742, 749, 756, 763, 770, 777, 784, 791, 798, 847, 875, 917, 973, 987, 1057, 1071, 1078, 1127, 1176, 1197, 1267, 1274, 1337, 1372, 1379

Offset: 1

Reinhard Zumkeller, Jul 21 2006

Paolo Xausa, Table of n, a(n) for n = 1..10000
Index entries for 10-automatic sequences.

Intersection of A008589 and A011537.

Cf. A121041, A011531, A121022, A121023, A121024, A121025, A121026, A121028, A121029, A121030, A121031, A121032, A121033, A121034, A121035, A121036, A121037, A121038, A121039, A121040.

Select[7*Range[500], MemberQ[IntegerDigits[#], 7] &] (* Paolo Xausa, Feb 25 2024 *)

is(n)=n%7==0 && setsearch(Set(digits(n)), 7) \\ Charles R Greathouse IV, Feb 12 2017

a(n) ~ 7n. - Charles R Greathouse IV, Feb 12 2017

Typo in comment fixed by Reinhard Zumkeller, May 01 2011

cp's OEIS Frontend

A355632 Irregular triangle T(n, k), n > 0, k = 1..A121041(n), read by rows; the n-th row contains in ascending order the divisors of n whose decimal expansions appear as substrings in the decimal expansion of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A011531 Numbers that contain a digit 1 in their decimal representation.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Haskell

Magma

Maple

Mathematica

PARI

Python

Scala

Formula

A121022 Even numbers containing a 2 in their decimal representation.

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

A121040 Multiples of 20 containing a 20 in their decimal representation.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A121025 Multiples of 5 containing a 5 in their decimal representation.

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Mathematica

PARI

Formula

Extensions

A121030 Multiples of 10 containing a 10 in their decimal representation.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A121032 Multiples of 12 containing a 12 in their decimal representation.

Views

Author

Keywords

Links

Crossrefs