A011535 -id:A011535 - cp's OEIS frontend

A043509 Numbers having exactly one 5 in base 10.

5, 15, 25, 35, 45, 50, 51, 52, 53, 54, 56, 57, 58, 59, 65, 75, 85, 95, 105, 115, 125, 135, 145, 150, 151, 152, 153, 154, 156, 157, 158, 159, 165, 175, 185, 195, 205, 215, 225, 235, 245, 250, 251, 252, 253, 254, 256, 257, 258, 259

Offset: 1

Clark Kimberling

Enrique Pérez Herrero, Table of n, a(n) for n = 1..2000
Index entries for 10-automatic sequences.

Cf. A043489, A043493, A043497, A043501, A043505, A043513, A043517, A043521, A043525, A011535.

Select[Range[9000],DigitCount[#,10,5]==1&] (* Enrique Pérez Herrero, Nov 29 2013 *)

a(n) ≍ n^k log n, with k = log 9/log 10 = 0.9542425... = A104139. - Charles R Greathouse IV, Nov 01 2022

A082834 Decimal expansion of Kempner series Sum_{k>=1, k has no digit 5 in base 10} 1/k.

Original entry on oeis.org

2, 1, 8, 3, 4, 6, 0, 0, 8, 1, 2, 2, 9, 6, 9, 1, 8, 1, 6, 3, 4, 0, 7, 2, 3, 5, 0, 4, 0, 6, 0, 9, 1, 8, 2, 7, 1, 7, 8, 4, 6, 5, 6, 7, 5, 1, 5, 0, 1, 3, 9, 1, 8, 2, 9, 1, 6, 7, 9, 3, 5, 9, 1, 8, 4, 2, 5, 0, 8, 6, 2, 6, 6, 8, 8, 2, 2, 9, 3, 8, 3, 5, 7, 7, 7, 2, 1, 3, 8, 3, 1, 9, 3, 2, 9, 2, 5, 4, 8, 8, 1, 3, 2, 4, 4

Offset: 2

Robert G. Wilson v, Apr 14 2003

Numbers with a digit 5 (A011535) have asymptotic density 1, i.e., here almost all terms are removed from the harmonic series, which makes convergence less surprising. See A082839 (the analog for digit 0) for more information about such so-called Kempner series. - M. F. Hasler, Jan 13 2020

			21.83460081229691816340723504060918271784656751501391829167935918... - _Robert G. Wilson v_, Jun 01 2009

Julian Havil, Gamma, Exploring Euler's Constant, Princeton University Press, Princeton and Oxford, 2003, page 34.

Robert Baillie, Sums of reciprocals of integers missing a given digit, Amer. Math. Monthly, 86 (1979), 372-374.
Robert Baillie, Summing the curious series of Kempner and Irwin arXiv:0806.4410 [math.CA], 2008-2015. [From _Robert G. Wilson v_, Jun 01 2009]
Wikipedia, Kempner series. [From _M. F. Hasler_, Jan 13 2020]
Wolfram Library Archive, KempnerSums.nb (8.6 KB) - Mathematica Notebook, Summing Kempner's Curious (Slowly-Convergent) Series. [From _Robert G. Wilson v_, Jun 01 2009]

Cf. A002387, A024101, A052413 (numbers with no '5'), A011535 (numbers with a '5').

Cf. A082830, A082831, A082832, A082833, A082835, A082836, A082837, A082838, A082839 (analog for digits 1, 2, ..., 9 and 0).

(* see the Mmca in Wolfram Library Archive. - Robert G. Wilson v, Jun 01 2009 *)

Equals Sum_{k in A052413\{0}} 1/k, where A052413 = numbers with no digit 5. - M. F. Hasler, Jan 15 2020

More terms from Robert G. Wilson v, Jun 01 2009

Minor edits by M. F. Hasler, Jan 13 2020

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

A062043 Positive numbers whose product of digits is 10 times their sum.

Original entry on oeis.org

459, 495, 549, 594, 945, 954, 1566, 1656, 1665, 2259, 2295, 2355, 2529, 2535, 2553, 2592, 2925, 2952, 3255, 3525, 3552, 5166, 5229, 5235, 5253, 5292, 5325, 5352, 5523, 5532, 5616, 5661, 5922, 6156, 6165, 6516, 6561, 6615, 6651, 9225, 9252, 9522

Offset: 1

Amarnath Murthy, Jun 28 2001

A subsequence of A011535 as each term must include the digit 5. - Chai Wah Wu, Dec 09 2015

			594 belongs to the sequence as (5*9*4)/(5+9+4) = 180/18 = 10.

Harry J. Smith, Table of n, a(n) for n = 1..500

Cf. A011540, A034710, A062034, A062035, A062036, A062382, A062037, A062384, A062040, A062041, A062045.

Cf. A011535, A124694.

filter:= proc(t) local L;
  L:= convert(t,base,10);
  convert(L,`*`) = 10*convert(L,`+`)
end proc:
select(filter, [$1..10^5]); # Robert Israel, Sep 11 2022

Select[Range[10000], Times @@ IntegerDigits[ # ] == 10 Plus @@ IntegerDigits[ # ] &] (* Tanya Khovanova, Dec 25 2006 *)

isok(n) = my(d=digits(n)); vecprod(d)==10*vecsum(d) \\ Mohammed Yaseen, Sep 11 2022

from math import prod
def ok(n): d = list(map(int, str(n))); return prod(d) == 10*sum(d)
print([k for k in range(1, 9999) if ok(k)]) # Michael S. Branicky, Sep 11 2022

More terms from Harvey P. Dale and Larry Reeves (larryr(AT)acm.org), Jul 06 2001

A293873 Numbers having '13' as substring of their digits.

Original entry on oeis.org

13, 113, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 213, 313, 413, 513, 613, 713, 813, 913, 1013, 1113, 1130, 1131, 1132, 1133, 1134, 1135, 1136, 1137, 1138, 1139, 1213, 1300, 1301, 1302, 1303, 1304, 1305, 1306, 1307, 1308, 1309, 1310, 1311, 1312, 1313, 1314, 1315

Offset: 1

M. F. Hasler, Oct 18 2017

Row 13 of A292690 and A293869. A121033 is the subsequence of multiples of 13.

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, 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.

Select[Range[1350],SequenceCount[IntegerDigits[#],{1,3}]>0&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 31 2017 *)

is_A293873 = has(n, p=13, 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

A043509 Numbers having exactly one 5 in base 10.

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A082834 Decimal expansion of Kempner series Sum_{k>=1, k has no digit 5 in base 10} 1/k.

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

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

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