A293880 -id:A293880 - cp's OEIS frontend

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

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

A293870 Numbers having '10' as substring of their digits.

Original entry on oeis.org

10, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 210, 310, 410, 510, 610, 710, 810, 910, 1000, 1001, 1002, 1003, 1004, 1005, 1006, 1007, 1008, 1009, 1010, 1011, 1012, 1013, 1014, 1015, 1016, 1017, 1018, 1019, 1020, 1021, 1022, 1023, 1024, 1025, 1026, 1027, 1028, 1029, 1030, 1031

Offset: 1

M. F. Hasler, Oct 18 2017

Row 10 of A292690 and A293869.

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. A293871, A293872, A293873, A293874, A293875, A293876, A293877, A293878, A293879, A293880: same for '11' - '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[1100],SequenceCount[IntegerDigits[#],{1,0}]>0&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 07 2019 *)

is_A293870 = has(n, p=10, m=10^#Str(p))=until(p>n\=10, n%m==p&&return(1))

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

A293874 Numbers having '14' as substring of their digits.

Original entry on oeis.org

14, 114, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 214, 314, 414, 514, 614, 714, 814, 914, 1014, 1114, 1140, 1141, 1142, 1143, 1144, 1145, 1146, 1147, 1148, 1149, 1214, 1314, 1400, 1401, 1402, 1403, 1404, 1405, 1406, 1407, 1408, 1409, 1410, 1411, 1412, 1413, 1414, 1415, 1416

Offset: 1

M. F. Hasler, Oct 18 2017

Row 14 of A292690 and A293869.

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

Cf. A292690, A293869. A121034 lists the terms which are divisible by 14.
Cf. A011540, A011531, A011532, A011533, A011534, A011535, A011536, A011537, A011538, A011539: analog for '0' - '9'.
Cf. A293870, A293871, A293872, A293873, 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.

Select[Range[2000], StringContainsQ[IntegerString[#], "14"] &] (* Paolo Xausa, Feb 25 2024 *)

is_A293874 = has(n, p=14, m=10^#Str(p))=until(p>n\=10, n%m==p&&return(1))

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

A293875 Numbers having '15' as substring of their digits.

Original entry on oeis.org

15, 115, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 215, 315, 415, 515, 615, 715, 815, 915, 1015, 1115, 1150, 1151, 1152, 1153, 1154, 1155, 1156, 1157, 1158, 1159, 1215, 1315, 1415, 1500, 1501, 1502, 1503, 1504, 1505, 1506, 1507, 1508, 1509, 1510, 1511, 1512, 1513, 1514, 1515

Offset: 1

M. F. Hasler, Oct 18 2017

Row 15 of A292690 and A293869. A121035 lists the terms which are divisible by 15.

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, 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.

Select[Range[2000], StringContainsQ[IntegerString[#], "15"] &] (* Paolo Xausa, Feb 25 2024 *)

is_A293875 = has(n, p=15, m=10^#Str(p))=until(p>n\=10, n%m==p&&return(1))

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

A293876 Numbers having '16' as substring of their digits / decimal expansion.

Original entry on oeis.org

16, 116, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 216, 316, 416, 516, 616, 716, 816, 916, 1016, 1116, 1160, 1161, 1162, 1163, 1164, 1165, 1166, 1167, 1168, 1169, 1216, 1316, 1416, 1516, 1600, 1601, 1602, 1603, 1604, 1605, 1606, 1607, 1608, 1609, 1610, 1611, 1612, 1613, 1614

Offset: 1

M. F. Hasler, Oct 18 2017

Row 16 of A292690 and A293869. A121036 lists the terms which are divisible by 16.

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, 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.

Select[Range[2000], StringContainsQ[IntegerString[#], "16"] &] (* Paolo Xausa, Feb 25 2024 *)

is_A293876 = has(n, p=16, m=10^#Str(p))=until(p>n\=10, n%m==p&&return(1))

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

A293878 Numbers having '18' as substring of their digits / decimal expansion.

Original entry on oeis.org

18, 118, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 218, 318, 418, 518, 618, 718, 818, 918, 1018, 1118, 1180, 1181, 1182, 1183, 1184, 1185, 1186, 1187, 1188, 1189, 1218, 1318, 1418, 1518, 1618, 1718, 1800, 1801, 1802, 1803, 1804, 1805, 1806, 1807, 1808, 1809, 1810, 1811, 1812

Offset: 1

M. F. Hasler, Oct 18 2017

Row 16 of A292690 and A293869. A121038 lists the terms which are divisible by 18.

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, A293877, 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[#], "18"] &] (* Paolo Xausa, Feb 25 2024 *)

is_A293878 = has(n, p=18, m=10^#Str(p))=until(p>n\=10, n%m==p&&return(1))

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

A293879 Numbers having '19' as substring of their digits.

Original entry on oeis.org

19, 119, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 219, 319, 419, 519, 619, 719, 819, 919, 1019, 1119, 1190, 1191, 1192, 1193, 1194, 1195, 1196, 1197, 1198, 1199, 1219, 1319, 1419, 1519, 1619, 1719, 1819, 1900, 1901, 1902, 1903, 1904, 1905, 1906, 1907, 1908, 1909, 1910, 1911

Offset: 1

M. F. Hasler, Oct 18 2017

Row 19 of A292690 and A293869. A121039 lists the terms which are divisible by 19.

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, A293877, A293878, 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],SequenceCount[IntegerDigits[#],{1,9}]>0&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 11 2019 *)

is_A293879 = has(n, p=19, 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

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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A293871 Numbers having 11 as substring of their digits.

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

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

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A293870 Numbers having '10' as substring of their digits.

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A293874 Numbers having '14' as substring of their digits.

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A293875 Numbers having '15' as substring of their digits.

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A293876 Numbers having '16' as substring of their digits / decimal expansion.

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