A011532 -id:A011532 - cp's OEIS frontend

A121022 Even numbers containing a 2 in their decimal representation.

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

A011533 Numbers that contain a 3.

Original entry on oeis.org

3, 13, 23, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 43, 53, 63, 73, 83, 93, 103, 113, 123, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 143, 153, 163, 173, 183, 193, 203, 213, 223, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 243, 253

Offset: 1

N. J. A. Sloane

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
James Grime and Brady Haran, 3 is everywhere, Numberphile video, 2012.
Index entries for 10-automatic sequences.

Complement: A052405.
Cf. A016189.
Numbers that contain a digit k: A011531 (k=1), A011532 (k=2), A011534 (k=4), A011535 (k=5), A011536 (k=6), A011537 (k=7), A011538 (k=8), A011539 (k=9), A011540 (k=0).

Filtered([1..260],n->3 in ListOfDigits(n)); # Muniru A Asiru, Feb 23 2019

a011533 n = a011533_list !! (n-1)
a011533_list = filter ((elem '3') . show) [0..]
-- Reinhard Zumkeller, Apr 10 2015

[n: n in [0..500] | 3 in Intseq(n)]; // Vincenzo Librandi, Jan 11 2016

M:= 3: # to get all terms of up to M digits
B:= {3}: A:= {3}:
for i from 2 to M do
   B:= map(t -> seq(10*t+j,j=0..9),B) union
      {seq(10*x+3,x=10^(i-2)..10^(i-1)-1)}:
   A:= A union B;
od:
sort(convert(A,list));# Robert Israel, Jan 11 2016

Select[Range[600] - 1, DigitCount[#, 10, 3]>0 &] (* Vincenzo Librandi, Jan 11 2016 *)

isok(n)=my(d=digits(n)); for (k=1, #d, if (d[k] == 3, return (1))); \\ Michel Marcus, Jan 11 2016

a(n) ~ n. - Charles R Greathouse IV, Aug 28 2012

For m >= 1, a(10^m - 9^m) = 10^m-7, a(10^m - 9^m + 1) = 10^m + 3. - Robert Israel, Jan 11 2016

A011534 Numbers that contain a 4.

Original entry on oeis.org

4, 14, 24, 34, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 54, 64, 74, 84, 94, 104, 114, 124, 134, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 154, 164, 174, 184, 194, 204, 214, 224, 234, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 254

Offset: 1

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 1..6878
Index entries for 10-automatic sequences.

Numbers that contain a digit k: A011531 (k=1), A011532 (k=2), A011533 (k=3), A011535 (k=5), A011536 (k=6), A011537 (k=7), A011538 (k=8), A011539 (k=9), A011540 (k=0).

Filtered([1..260],n->4 in ListOfDigits(n)); # Muniru A Asiru, Feb 23 2019

Select[Range[200], DigitCount[#, 10, 4]>0 &] (* Vincenzo Librandi, Feb 14 2017 *)

is(n)=!!setsearch(Set(digits(n)), 4) \\ Charles R Greathouse IV, Feb 12 2017

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

A256786 Numbers which are divisible by prime(d) for all digits d in their decimal representation.

Original entry on oeis.org

12, 14, 42, 55, 154, 222, 228, 714, 1122, 1196, 1212, 1414, 2112, 2142, 2262, 3355, 4144, 4242, 5335, 5544, 5555, 6162, 9499, 11112, 11144, 11214, 11424, 11466, 11622, 11818, 11914, 12222, 12882, 14112, 15554, 16666, 21216, 21222, 21252, 21888, 22122, 22212

Offset: 1

Eric Angelini and Reinhard Zumkeller, Apr 10 2015

All terms are zerofree, cf. A052382;
there is no term containing digits 1 and 3 simultaneously;
a(n) contains at least one digit 1 iff a(n) is even, cf. A011531, A005843;
a(n) contains at least one digit 2 iff a(n) mod 3 = 0, cf. A011532, A008585;
a(n) contains at least one digit 3 iff a(n) mod 10 = 5, cf. A011533, A017329;
A020639(a(n)) <= 23.
The equivalent in base 2 is the empty sequence, in base 3 it is A191681\{0}; see A256874-A256879 for the base 4 - base 9 variant, and A256870 for a variant where digits 0 are allowed but divisibility by prime(d+1) is required instead. - M. F. Hasler, Apr 11 2015

			Smallest terms containing the nonzero decimal digits:
.  d | prime(d) |  n | a(n)
. ---+----------+--------------------------
.  1 |       2  |  1 |   12 = 2^2 * 3
.  2 |       3  |  1 |   12 = 2^2 * 3
.  3 |       5  | 16 | 3355 = 5 * 11 * 61
.  4 |       7  |  2 |   14 = 2 * 7
.  5 |      11  |  4 |   55 = 5 * 11
.  6 |      13  | 10 | 1196 = 2^2 * 13 * 23
.  7 |      17  |  8 |  714 = 2 * 3 * 7 * 17
.  8 |      19  |  7 |  228 = 2^2 * 3 * 19
.  9 |      23  | 10 | 1196 = 2^2 * 13 * 23 .

Lars Blomberg and Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Éric Angelini, Divisibility by primes, SeqFan list, Apr 10 2015.
Index entries for 10-automatic sequences.

Cf. A000040, A005843, A008585, A011531, A011532, A011533, A017329, A020639, A052382, A256874-A256879, A256882-A256884, A256865-A256870.

a256786 n = a256786_list !! (n-1)
a256786_list = filter f a052382_list where
   f x = g x where
     g z = z == 0 || x `mod` a000040 d == 0 && g z'
           where (z', d) = divMod z 10

Select[Range@22222,FreeQ[IntegerDigits[#],0]&&Total[Mod[#,Prime[IntegerDigits[#]]]]==0&] (* Ivan N. Ianakiev, Apr 11 2015 *)

is_A256786(n)=!for(i=1,#d=Set(digits(n)),(!d[i]||n%prime(d[i]))&&return) \\ M. F. Hasler, Apr 11 2015

primes = [1, 2, 3, 5, 7, 11, 13, 17, 19, 23]
def ok(n):
    s = str(n)
    return "0" not in s and all(n%primes[int(d)] == 0 for d in s)
print([k for k in range(22213) if ok(k)]) # Michael S. Branicky, Dec 14 2021

A217394 Numbers starting with 2.

Original entry on oeis.org

2, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242

Offset: 1

Jeremy Gardiner, Oct 02 2012

The lower and upper asymptotic densities of this sequence are 1/18 and 10/27, respectively. - Amiram Eldar, Feb 27 2021

Jeremy Gardiner, Table of n, a(n) for n = 1..1111
Index entries for 10-automatic sequences.

Cf. A006092, A011532.
Subsequences include: A045708, A045726, A077327, A077678, A106412, A106422.
Cf. A131835, A217395, A217397, A217398, A217399, A217400, A217401, A217402.

Select[Range[300], IntegerDigits[#][[1]] == 2 &] (* T. D. Noe, Oct 02 2012 *)

def agen():
  yield 2
  digits, adder = 1, 20
  while True:
    for i in range(10**digits): yield adder + i
    digits, adder = digits+1, adder*10
g = agen()
print([next(g) for i in range(54)]) # Michael S. Branicky, Feb 20 2021

def A217394(n): return n+(17*10**(len(str(9*n-8))-1))//9 # Chai Wah Wu, Dec 07 2024

a(n) = n + (17*10^floor(log_10(9*n-8))-8)/9. - Alan Michael Gómez Calderón, May 15 2023

A293869 Square array whose n-th row lists all numbers having n as a substring, n >= 1; read by falling antidiagonals.

Original entry on oeis.org

1, 10, 2, 11, 12, 3, 12, 20, 13, 4, 13, 21, 23, 14, 5, 14, 22, 30, 24, 15, 6, 15, 23, 31, 34, 25, 16, 7, 16, 24, 32, 40, 35, 26, 17, 8, 17, 25, 33, 41, 45, 36, 27, 18, 9, 18, 26, 34, 42, 50, 46, 37, 28, 19, 10, 19, 27, 35, 43, 51, 56, 47, 38, 29, 100, 11

Offset: 1

M. F. Hasler, Oct 18 2017

			The array starts:
   [ 1  10  11  12  13  14  15  16  17  18  19  21  31 ...] = A011531
   [ 2  12  20  21  22  23  24  25  26  27  28  29  32 ...] = A011532
   [ 3  13  23  30  31  32  33  34  35  36  37  38  39 ...] = A011533
   [ 4  14  24  34  40  41  42  43  44  45  46  47  48 ...] = A011534
   [ 5  15  25  35  45  50  51  52  53  54  55  56  57 ...] = A011535
   [ 6  16  26  36  46  56  60  61  62  63  64  65  66 ...] = A011536
   [ 7  17  27  37  47  57  67  70  71  72  73  74  75 ...] = A011537
   [ 8  18  28  38  48  58  68  78  80  81  82  83  84 ...] = A011538
   [ 9  19  29  39  49  59  69  79  89  90  91  92  93 ...] = A011539
   [10 100 101 102 103 104 105 106 107 108 109 110 210 ...] = A293870
   [11 110 111 112 113 114 115 116 117 118 119 211 311 ...] = A293871
   [12 112 120 121 122 123 124 125 126 127 128 129 212 ...] = A293872
   [   ...             ...             ...             ...]

Rémy Sigrist, Table of n, a(n) for n = 1..5050
Rémy Sigrist, Perl program for A293869

Cf. A072484, A292690 (variant starting with row 0).
Cf. A011540, A011531, A011532, A011533, A011534, A011535, A011536, A011537, A011538, A011539, A293870, A293871, A293872.
Cf. A121041, A121022, A121023, A121024, A121025, A121026, A121027, A121028, A121029, A121030, A121031, A121032, A121033, A121034, A121035, A121036, A121037, A121038, A121039, A121040.
Cf. A179239, A261370.
Cf. A292451, A292731 (both partially coincide with row 11, but no inclusion relation holds).

Block[{d = 15, q, a, s}, a = Table[q = n-1; s = IntegerString[n]; Table[While[StringFreeQ[IntegerString[++q], s]]; q, d-n+1], {n, d}]; Table[a[[n, k-n+1]], {k, d}, {n, k}]] (* Paolo Xausa, Mar 01 2024 *)

has=(n,p,m=10^#Str(p))->until(p>n\=10,n%m==p&&return(1))
Mat(vectorv(12,n,a=[];for(k=n,oo,has(k,n)||next;a=concat(a,k);#a>12&&break);a))

Perl
```
See Links section.
```

T(n, k) = A072484(n, k) for any n > 0 and k = 1..n. - Rémy Sigrist, Jan 29 2021

A292690 Square array whose n-th row lists all numbers having n as a substring, read by falling antidiagonals, n >= 0.

Original entry on oeis.org

0, 10, 1, 20, 10, 2, 30, 11, 12, 3, 40, 12, 20, 13, 4, 50, 13, 21, 23, 14, 5, 60, 14, 22, 30, 24, 15, 6, 70, 15, 23, 31, 34, 25, 16, 7, 80, 16, 24, 32, 40, 35, 26, 17, 8, 90, 17, 25, 33, 41, 45, 36, 27, 18, 9, 100, 18, 26, 34, 42, 50, 46, 37, 28, 19, 10

Offset: 0

M. F. Hasler, Oct 18 2017

This array starts with row 0, see A293869 for the variant which starts with row 1.

			The array starts:
   [ 0  10  20  30  40  50  60  70  80  90 100 101 102 ...] = A011540
   [ 1  10  11  12  13  14  15  16  17  18  19  21  31 ...] = A011531
   [ 2  12  20  21  22  23  24  25  26  27  28  29  32 ...] = A011532
   [ 3  13  23  30  31  32  33  34  35  36  37  38  39 ...] = A011533
   [ 4  14  24  34  40  41  42  43  44  45  46  47  48 ...] = A011534
   [ 5  15  25  35  45  50  51  52  53  54  55  56  57 ...] = A011535
   [ 6  16  26  36  46  56  60  61  62  63  64  65  66 ...] = A011536
   [ 7  17  27  37  47  57  67  70  71  72  73  74  75 ...] = A011537
   [ 8  18  28  38  48  58  68  78  80  81  82  83  84 ...] = A011538
   [ 9  19  29  39  49  59  69  79  89  90  91  92  93 ...] = A011539
   [10 100 101 102 103 104 105 106 107 108 109 110 210 ...] = A293870
   [11 110 111 112 113 114 115 116 117 118 119 211 311 ...] = A293871
   [12 112 120 121 122 123 124 125 126 127 128 129 212 ...] = A293872
   [   ...             ...             ...             ...]

Paolo Xausa, Table of n, a(n) for n = 0..11324 (first 150 antidiagonals).

Cf. A293869.
Cf. A011540, A011531, A011532, A011533, A011534, A011535, A011536, A011537, A011538, A011539: rows 0 - 9.
Cf.  A293870, A293871, A293872, A293873, A293874, A293875, A293876, A293877, A293878, A293879, A293880: rows 10 .. 20.
Cf. A121041, A121022, A121023, A121024, A121025, A121026, A121027, A121028, A121029, A121030, A121031, A121032, A121033, A121034, A121035, A121036, A121037, A121038, A121039, A121040: subsequences of the above, only multiples of the pattern p.

Block[{d = 15, q, a, s}, a = Table[q = n-1; s = IntegerString[n]; Table[While[StringFreeQ[IntegerString[++q], s]]; q, d-n], {n, 0, d-1}]; Table[a[[n+1, k-n]], {k, d}, {n, 0, k-1}]] (* Paolo Xausa, Mar 01 2024 *)

has(n,p,m=10^#Str(p))=until(p+!p>n\=10,n%m==p&&return(1))
Mat(vectorv(12,n,a=[];for(k=n--,oo,has(k,n)||next;a=concat(a,k);#a>12&&break);a))
for(i=1,11,for(j=1,i,print1(%[j,i-j+1]","))) \\ Read by antidiagonals

A293871 Numbers having 11 as substring of their digits.

Original entry on oeis.org

11, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 211, 311, 411, 511, 611, 711, 811, 911, 1011, 1100, 1101, 1102, 1103, 1104, 1105, 1106, 1107, 1108, 1109, 1110, 1111, 1112, 1113, 1114, 1115, 1116, 1117, 1118, 1119, 1120, 1121, 1122, 1123, 1124, 1125, 1126, 1127, 1128, 1129, 1130, 1131

Offset: 1

M. F. Hasler, Oct 18 2017

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

Row 11 of A292690 and A293869.
Cf. A292451, A292731 (both partially coincide with this sequence, but no inclusion relation holds).
Cf. A011540, A011531, A011532, A011533, A011534, A011535, A011536, A011537, A011538, A011539: analog for substrings '0' through '9'.
Cf. A293870, A293872, A293873, A293874, A293875, A293876, A293877, A293878, A293879, A293880: same for substrings '10' - '20'.
Cf. A121031: subsequence of terms divisible by 11.
Numbers divisible by k and having k as a substring: A121022 (2), A121023 (3), A121024 (4), A121025 (5), A121026 (6), A121027 (7), A121028 (8), A121029 (9), A121030 (10), A121031 (11), A121032 (12), A121033 (13), A121034 (14), A121035 (15), A121036 (16), A121037 (17), A121038 (18), A121039 (19), A121040 (20).
Cf. A121041.

Select[Range[2000], StringContainsQ[IntegerString[#], "11"] &] (* Paolo Xausa, Feb 25 2024 *)

is_A293871 = has(n,p=11,m=10^#Str(p))=until(p>n\=10,n%m==p&&return(1))

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

A293872 Numbers having '12' as a substring of their digits.

Original entry on oeis.org

12, 112, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 212, 312, 412, 512, 612, 712, 812, 912, 1012, 1112, 1120, 1121, 1122, 1123, 1124, 1125, 1126, 1127, 1128, 1129, 1200, 1201, 1202, 1203, 1204, 1205, 1206, 1207, 1208, 1209, 1210, 1211, 1212, 1213, 1214, 1215, 1216, 1217, 1218

Offset: 1

M. F. Hasler, Oct 18 2017

Row 12 of A292690 and A293869. A121032 is the subsequence of multiples of 12.

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

Cf. A292690, A293869.
Cf. A011540, A011531, A011532, A011533, A011534, A011535, A011536, A011537, A011538, A011539: analog for '0' - '9'.
Cf. A293870, A293871, A293873, A293874, A293875, A293876, A293877, A293878, A293879, A293880: same for '10' - '20'.
Cf. A121041, A121022, A121023, A121024, A121025, A121026, A121027, A121028, A121029, A121030, A121031, A121032, A121033, A121034, A121035, A121036, A121037, A121038, A121039, A121040: subsequences of the above, containing only multiples of the pattern p.

f:= proc(d) local i,x,y;
  sort(convert({seq(seq(seq(x+10^i*12+10^(i+2)*y, y=10^(d-3-i)..10^(d-2-i)-1),x=0..10^i-1),i=0..d-3),
seq(12*10^(d-2)+x,x=0..10^(d-2)-1)},list))
end proc:
seq(op(f(d)),d=2..4); # Robert Israel, Nov 20 2017

Select[Range@ 1220, SequenceCount[IntegerDigits[#], {1, 2}] > 0 &] (* Michael De Vlieger, Nov 20 2017 *)

is_A293872 = has(n, p=12, m=10^#Str(p))=until(p>n\=10, n%m==p&&return(1))

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

A293877 Numbers having '17' as substring of their digits / decimal expansion.

Original entry on oeis.org

17, 117, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 217, 317, 417, 517, 617, 717, 817, 917, 1017, 1117, 1170, 1171, 1172, 1173, 1174, 1175, 1176, 1177, 1178, 1179, 1217, 1317, 1417, 1517, 1617, 1700, 1701, 1702, 1703, 1704, 1705, 1706, 1707, 1708, 1709, 1710, 1711, 1712, 1713

Offset: 1

M. F. Hasler, Oct 18 2017

Row 17 of A292690 and A293869. A121037 lists the terms which are divisible by 17.

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

Cf. A292690, A293869.
Cf. A011540, A011531, A011532, A011533, A011534, A011535, A011536, A011537, A011538, A011539: analog for '0' - '9'.
Cf. A293870, A293871, A293872, A293873, A293874, A293875, A293876, A293878, A293879, A293880: same for '10' - '20'.
Cf. A121041, A121022, A121023, A121024, A121025, A121026, A121027, A121028, A121029, A121030, A121031, A121032, A121033, A121034, A121035, A121036, A121037, A121038, A121039, A121040: subsequences of the above, containing only multiples of the pattern p.

Select[Range[2000], StringContainsQ[IntegerString[#], "17"] &] (* Paolo Xausa, Feb 25 2024 *)

is_A293877 = has(n, p=17, m=10^#Str(p))=until(p>n\=10, n%m==p&&return(1))

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

cp's OEIS Frontend

A121022 Even numbers containing a 2 in their decimal representation.

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A011533 Numbers that contain a 3.

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A011534 Numbers that contain a 4.

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A256786 Numbers which are divisible by prime(d) for all digits d in their decimal representation.

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A217394 Numbers starting with 2.

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A293869 Square array whose n-th row lists all numbers having n as a substring, n >= 1; read by falling antidiagonals.

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A292690 Square array whose n-th row lists all numbers having n as a substring, read by falling antidiagonals, n >= 0.

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