cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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

Views

Author

M. F. Hasler, Oct 18 2017

Keywords

Comments

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

Links

Crossrefs

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.

Programs

  • Mathematica
    Select[Range[2000], StringContainsQ[IntegerString[#], "16"] &] (* Paolo Xausa, Feb 25 2024 *)
  • PARI
    is_A293876 = has(n, p=16, m=10^#Str(p))=until(p>n\=10, n%m==p&&return(1))

Formula

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

Views

Author

M. F. Hasler, Oct 18 2017

Keywords

Comments

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

Links

Crossrefs

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.

Programs

  • Mathematica
    Select[Range[2000], StringContainsQ[IntegerString[#], "18"] &] (* Paolo Xausa, Feb 25 2024 *)
  • PARI
    is_A293878 = has(n, p=18, m=10^#Str(p))=until(p>n\=10, n%m==p&&return(1))

Formula

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

Views

Author

M. F. Hasler, Oct 18 2017

Keywords

Comments

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

Links

Crossrefs

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.

Programs

  • Mathematica
    Select[Range[2000],SequenceCount[IntegerDigits[#],{1,9}]>0&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 11 2019 *)
  • PARI
    is_A293879 = has(n, p=19, m=10^#Str(p))=until(p>n\=10, n%m==p&&return(1))

Formula

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

A043525 Numbers having one 9 in base 10.

Original entry on oeis.org

9, 19, 29, 39, 49, 59, 69, 79, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 109, 119, 129, 139, 149, 159, 169, 179, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 209, 219, 229, 239, 249, 259, 269, 279, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 309, 319
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A043489, A043493, A043497, A043501, A043505, A043509, A043513, A043517, A043521.
Cf. A011539, A140502.
Subsequence of A011539.

Programs

  • Mathematica
    Select[Range[300],DigitCount[#,10,9]==1&] (* Harvey P. Dale, Jan 19 2013 *)
  • Python
    def ok(n): return str(n).count('9') == 1
print(list(filter(ok, range(320)))) # Michael S. Branicky, Sep 19 2021

Formula

Sum_{n>=1} 1/a(n) = A140502. - Amiram Eldar, Nov 14 2020

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

Views

Author

M. F. Hasler, Oct 18 2017

Keywords

Comments

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

Links

Crossrefs

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.

Programs

  • Mathematica
    Select[Range[1350],SequenceCount[IntegerDigits[#],{1,3}]>0&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 31 2017 *)
  • PARI
    is_A293873 = has(n, p=13, m=10^#Str(p))=until(p>n\=10, n%m==p&&return(1))

Formula

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

A293880 Numbers having '20' as substring of their digits.

Original entry on oeis.org

20, 120, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 220, 320, 420, 520, 620, 720, 820, 920, 1020, 1120, 1200, 1201, 1202, 1203, 1204, 1205, 1206, 1207, 1208, 1209, 1220, 1320, 1420, 1520, 1620, 1720, 1820, 1920, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010
Offset: 1

Views

Author

M. F. Hasler, Oct 18 2017

Keywords

Comments

Row 20 of A292690 and A293869. A121040 lists the terms which are divisible by 19.

Links

Crossrefs

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, A293879: same for '10' - '19'.
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.

Programs

  • Mathematica
    Select[Range[2100],SequenceCount[IntegerDigits[#],{2,0}]>0&] (* Harvey P. Dale, Jul 25 2021 *)
  • PARI
    is_A293880 = has(n, p=20, m=10^#Str(p))=until(p>n\=10, n%m==p&&return(1))

Formula

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

A068505 Decimal representation of n interpreted in base b+1, where b=A054055(n) is the largest digit in decimal representation of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 3, 5, 7, 9, 11, 13, 15, 17, 19, 6, 7, 8, 11, 14, 17, 20, 23, 26, 29, 12, 13, 14, 15, 19, 23, 27, 31, 35, 39, 20, 21, 22, 23, 24, 29, 34, 39, 44, 49, 30, 31, 32, 33, 34, 35, 41, 47, 53, 59, 42, 43, 44, 45, 46, 47, 48, 55, 62, 69, 56, 57, 58, 59, 60, 61
Offset: 1

Views

Author

Reinhard Zumkeller, Mar 11 2002, Feb 23 2008

Keywords

Comments

a(n) = n iff n < 10 OR n is a "9ish number": a(A011539(n)) = A011539(n). - Reinhard Zumkeller, Dec 29 2011

Examples

			a(20)=2*3^1+0*1=6, a(21)=2*3^1+1*1=7, a(22)=2*3^1+2*1=8,
a(23)=2*4^1+3*1=11, a(24)=2*5^1+4*1=14, a(25)=2*6^1+5*1=17,
a(26)=2*7^1+6*1=20, a(27)=2*8^1+7*1=23, a(28)=2*9^1+8*1=26,
a(29)=2*10^1+9*1=29, a(30)=3*4^1+0*1=12, a(31)=3*4^1+1*1=13.

Links

Crossrefs

Cf. A031298.

Programs

  • Haskell
    a068505 n = foldr (\d v -> v * b + d) 0 dds where
b = maximum dds + 1
dds = a031298_row n
-- Reinhard Zumkeller, Feb 17 2013, Dec 29 2011
  • Maple
    f:= proc(n) local b,L,i;
L:= convert(n,base,10);
b:= max(L);
add(L[i]*(b+1)^(i-1),i=1..nops(L));
end proc:
map(f, [$1..100]); # Robert Israel, Feb 02 2016
  • Mathematica
    a[n_] := (id = IntegerDigits[n] // Reverse; b = Max[id]+1; id.b^Range[0, Length[id]-1]); Table[a[n], {n, 1, 75}] (* Jean-François Alcover, May 15 2013 *)
Table[FromDigits[IntegerDigits[n],Max[IntegerDigits[n]+1]],{n,80}] (* Harvey P. Dale, Dec 02 2015 *)
  • PARI
    a(n)=my(d = digits(n), b = vecmax(d)); subst(Pol(d), x, b+1); \\ Michel Marcus, Feb 12 2016

Extensions

Definition clarified and comment corrected by Martin Büttner, Feb 02 2016

A088924 Number of "9ish numbers" with n digits.

Original entry on oeis.org

1, 18, 252, 3168, 37512, 427608, 4748472, 51736248, 555626232, 5900636088, 62105724792, 648951523128, 6740563708152, 69665073373368, 716985660360312, 7352870943242808, 75175838489185272, 766582546402667448
Offset: 1

Views

Author

Marc LeBrun, Oct 23 2003

Keywords

Comments

First difference of A016189. ("9" can be replaced by any other nonzero digit, however only the 9ish numbers are closed under lunar multiplication.)
See A257285 - A257289 for first differences of 5^n-4^n, ..., 9^n-8^n. These also give the number of n-digit numbers whose largest digit is 5, 6, 7, 8, respectively. - M. F. Hasler, May 04 2015

Examples

			a(2) = 18 because 19, 29, 39, 49, 59, 69, 79, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98 and 99 are the eighteen two-digit 9ish numbers.

Links

Crossrefs

Cf. A016189, A011539, A257285, A257286, A257287, A257288, A257289.

Programs

Formula

a(n) = 9*10^(n-1) - 8*9^(n-1).
G.f.: x*(1 - x)/(1 - 19*x + 90*x^2). - Bobby Milazzo, May 02 2014
a(n) = 19*a(n-1) - 90*a(n-2). - Vincenzo Librandi, May 04 2015
E.g.f.: (81*exp(10*x) - 80*exp(9*x) - 1)/90. - Stefano Spezia, Nov 16 2023

A257226 Numbers that have at least one divisor containing the digit 9 in base 10.

Original entry on oeis.org

9, 18, 19, 27, 29, 36, 38, 39, 45, 49, 54, 57, 58, 59, 63, 69, 72, 76, 78, 79, 81, 87, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 108, 109, 114, 116, 117, 118, 119, 126, 129, 133, 135, 138, 139, 144, 145, 147, 149, 152, 153, 156, 158, 159, 162, 169, 171, 174
Offset: 1

Views

Author

Jaroslav Krizek, May 29 2015

Keywords

Comments

Numbers k whose concatenation of divisors A037278(k), A176558(k), A243360(k) or A256824(k) contains a digit 9.
A011539 (numbers that contain a 9) is a subsequence.

Examples

			18 is in sequence because the list of divisors of 18: (1, 2, 3, 6, 9, 18) contains digit 9.

Crossrefs

Cf. A037278, A176558, A243360, A256824.
Cf. similar sequences with another digit: A209932 (0), A000027 (1), A257219 (2), A257220 (3), A257221 (4), A257222 (5), A257223 (6), A257224 (7), A257225 (8).

Programs

  • Magma
    [n: n in [1..1000] | [9] subset Setseq(Set(Sort(&cat[Intseq(d): d in Divisors(n)])))];
  • Mathematica
    Select[Range@108, Part[Plus @@ DigitCount@ Divisors@ #, 9] > 0 &] (* after Michael De Vlieger *)
  • PARI
    is(n)=fordiv(n, d, if(setsearch(Set(digits(d)), 9), return(1))); 0 \\ after Charles R Greathouse IV
  • Python
    from itertools import count, islice
from sympy import divisors
def A257226_gen(): return filter(lambda n:any('9' in str(d) for d in divisors(n,generator=True)),count(1))
A257226_list = list(islice(A257226_gen(),20)) # Chai Wah Wu, Dec 27 2021

Formula

a(n) ~ n.

A175688 Numbers k with property that arithmetic mean of its digits is both an integer and one of the digits of k.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 102, 111, 120, 123, 132, 135, 147, 153, 159, 174, 195, 201, 204, 210, 213, 222, 231, 234, 240, 243, 246, 258, 264, 285, 306, 312, 315, 321, 324, 333, 342, 345, 351, 354, 357, 360, 369, 375, 396, 402
Offset: 1

Views

Author

Claudio Meller, Aug 09 2010

Keywords

Comments

Subsequence of A061383.
A180160(a(n)) = 0. - Reinhard Zumkeller, Aug 15 2010

Examples

			135 is in the list because (1+3+5)/3 = 3 and 3 is a digit of 135.

Links

Crossrefs

Cf. A007953, A055642, A004426, A004427; A011531, A011532, A011533, A011534, A011535, A011536, A011537, A011538, A011539.
Cf. A052018. - Reinhard Zumkeller, Aug 15 2010

Programs

  • Haskell
    a175688 n = a175688_list !! (n-1)
a175688_list = filter f [0..] where
   f x = m == 0 && ("0123456789" !! avg) `elem` show x
         where (avg, m) = divMod (a007953 x) (a055642 x)
-- Reinhard Zumkeller, Jun 18 2013
  • Mathematica
    idQ[n_]:=Module[{idn=IntegerDigits[n],m},m=Mean[idn];IntegerQ[m] && MemberQ[idn,m]]; Select[Range[0,500],idQ] (* Harvey P. Dale, Jun 10 2011 *)

Extensions

Edited by Reinhard Zumkeller, Aug 13 2010
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