A011531 -id:A011531 - cp's OEIS frontend

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

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

A038770 Numbers divisible by at least one of their digits.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 28, 30, 31, 32, 33, 35, 36, 39, 40, 41, 42, 44, 45, 48, 50, 51, 52, 55, 60, 61, 62, 63, 64, 65, 66, 70, 71, 72, 75, 77, 80, 81, 82, 84, 85, 88, 90, 91, 92, 93, 95, 96, 99, 100, 101, 102

Offset: 1

Henry Bottomley, May 04 2000

A038769(a(n)) > 0; complement of A038772.

The decimal digit strings of this sequence are a regular language, since it is the union of A011531 and A121022 .. A121029 which are likewise regular languages. Some computer state machine manipulation for this union shows a minimum deterministic finite automaton (DFA) matching the digit strings of this sequence has 11561 states. Reversing strings (so least significant digit first) reduces to 1699 states, or reverse and allow high 0's (which become trailing 0's due to the reverse) reduces to 1424 states. The latter are tractable sizes for the linear recurrence in A327560. - Kevin Ryde, Dec 04 2019

			35 is included because 5 is a divisor of 35, but 37 is not included because neither 3 nor 7 is a divisor of 37.

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

Cf. A327560 (counts), A038772 (complement), A034709, A034837, A038769.

a038770 n = a038770_list !! (n-1)
a038770_list = filter f [1..] where
   f u = g u where
     g v = v > 0 && (((d == 0 || r > 0) && g v') || r == 0)
           where (v',d) = divMod v 10; r = mod u d
-- Reinhard Zumkeller, Jul 30 2015, Jun 19 2011

Select[Range[120],MemberQ[Divisible[#,Cases[IntegerDigits[#],Except[0]]], True]&] (* Harvey P. Dale, Jun 20 2011 *)
Select[Range[120],AnyTrue[#/DeleteCases[IntegerDigits[#],0],IntegerQ]&] (* Harvey P. Dale, Mar 29 2024 *)

is(n)=my(v=vecsort(eval(Vec(Str(n))),,8));for(i=if(v[1],1,2),#v,if(n%v[i]==0,return(1)));0 \\ Charles R Greathouse IV, Jul 22 2011

def ok(n): return any(n%int(d) == 0 for d in str(n) if d != '0')
print(list(filter(ok, range(1, 103)))) # Michael S. Branicky, May 20 2021

a(n) ~ n. - Charles R Greathouse IV, Jul 22 2011

A118950 Numbers containing at least one prime digit.

Original entry on oeis.org

2, 3, 5, 7, 12, 13, 15, 17, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 42, 43, 45, 47, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 62, 63, 65, 67, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 82, 83, 85, 87, 92, 93, 95, 97, 102, 103, 105, 107, 112

Offset: 1

Rick L. Shepherd, May 06 2006

A193238(a(n)) > 0; complement of A084984; A092620, A092624 and A092625 are subsequences. - Reinhard Zumkeller, Jul 19 2011

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

Cf. A118951, A092620, A084984, A011540, A011531, A046034, A179336 (primes).

a118950 n = a118950_list !! (n-1)
a118950_list = filter (any (`elem` "2357") . show ) [0..]
-- Reinhard Zumkeller, Jul 19 2011

Select[Range[150],AnyTrue[IntegerDigits[#],PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 19 2018 *)

is(n)=!!#select(isprime, digits(n)) \\ Charles R Greathouse IV, Sep 15 2015

a(n) = n + O(n^k) with k = log 6/log 10 = 0.77815.... - Charles R Greathouse IV, Sep 15 2015

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

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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A038770 Numbers divisible by at least one of their digits.

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A118950 Numbers containing at least one prime digit.

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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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