A043521 -id:A043521 - cp's OEIS frontend

A011538 Numbers that contain an 8.

8, 18, 28, 38, 48, 58, 68, 78, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 98, 108, 118, 128, 138, 148, 158, 168, 178, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 198, 208, 218, 228, 238, 248, 258, 268, 278, 280, 281, 282, 283, 284, 285, 286, 287, 288, 289

Offset: 1

N. J. A. Sloane

In China, numbers with many 8 digits are considered auspicious because the character for 8 (八) sounds like the character for wealth (發). - Charles R Greathouse IV, Aug 28 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..3439
Index entries for 10-automatic sequences.

Supersequence of A043521.

Filtered([1..290],n->8 in ListOfDigits(n)); # Muniru A Asiru, Feb 24 2019

[n: n in [0..400] | 8 in Intseq(n) ]; // Vincenzo Librandi, Feb 15 2017

Select[Range[400], DigitCount[#, 10, 8]>0 &] (* Vincenzo Librandi, Feb 15 2017 *)

is(n)=!!setsearch(Set(digits(n)), 8) \\ Charles R Greathouse IV, Feb 12 2017

def ok(n): return '8' in str(n)
print(list(filter(ok, range(290)))) # Michael S. Branicky, Aug 18 2021

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

A043489 Numbers having one 0 in base 10.

Original entry on oeis.org

0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 120, 130, 140, 150, 160, 170, 180, 190, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 220, 230, 240, 250, 260, 270, 280, 290, 301, 302, 303

Offset: 1

Clark Kimberling

From Hieronymus Fischer, May 28 2014: (Start)
Inversion:
Given a term m, the index n such that a(n) = m can be calculated by the following procedure [see Prog section with an implementation in Smalltalk]. With k := floor(log_10(m)), z = digit position of the '0' in m counted from the right (starting with 0).
Case 1: A043489_inverse(m) = 1 + Sum_{j=1..k} A052382_inverse(floor(m/10^j))*9^(j-1), if z = 0.
Case 2: A043489_inverse(m) = 1 + A043489_inverse(m - c - m mod 10^z) + A052382_inverse(m mod 10^z)) - (9^z - 1)/8, if z > 0, where c := 1, if the digit at position z+1 of m is ‘1’ and k > z + 1, otherwise c := 10.
Example 1: m = 990, k = 2, z = 0 (Case 1), A043489_inverse(990) = 1 + A052382_inverse(99))*1 + A052382_inverse(9))*9 = 1 + 90 + 81 = 172.
Example 2: m = 1099, k = 3, z = 2 (Case 2), A043489_inverse(1099) = 1 + A043489_inverse(990) + A052382_inverse(99)) - 10 = 1 +  A043489_inverse(990) + 80 = 1 + 172 + 80 = 253.
(End)

			a(10^1)= 90.
a(10^2)= 590.
a(10^3)= 4190.
a(10^4)= 35209.
a(10^5)= 308949.
a(10^6)= 2901559.
a(10^7)= 27250269.
a(10^8)= 263280979.
a(10^9)= 2591064889.
a(10^10)= 25822705899.
a(10^20)= 366116331598324670219.
a(10^50)= 3.7349122484477433715662812...*10^51
a(10^100)= 4.4588697999177752943575344...*10^103.
a(10^1000)= 5.5729817962143143812258616...*10^1045.
[Examples by _Hieronymus Fischer_, May 28 2014]

Enrique Pérez Herrero and Hieronymus Fischer [terms 1..2000 from Enrique Pérez Herrero], Table of n, a(n) for n = 1..10000
Index entries for 10-automatic sequences.

Cf. A043493, A043497, A043501, A043505, A043509, A043513, A043517, A043521, A043525, A011540, A052382.

Select[Range[0,9000],DigitCount[#,10,0]==1&] (* Enrique Pérez Herrero, Nov 29 2013 *)

is(n)=#select(d->d==0, digits(n))==1 \\ Charles R Greathouse IV, Oct 06 2016

A043489_nextTerm
  "Answers the minimal number > m which contains exactly 1 zero digit (in base 10), where m is the receiver.
  Usage: a(n) A043489_nextTerm
  Answer: a(n+1)"
  | d d0 s n p |
  n := self.
  p := 1.
  s := n.
  (d0 := n // p \\ 10) = 0
     ifTrue:
          [p := 10 * p.
          s := s + 1].
  [(d := n // p \\ 10) = 9] whileTrue:
          [s := s - (8 * p).
          p := 10 * p].
  (d = 0 or: [d0 = 0]) ifTrue: [s := s - (p // 10)].
  ^s + p
[by Hieronymus Fischer, May 28 2014]
------------------

A043489
"Answers the n-th number such that number of 0's in base 10 is 1, where n is the receiver. Uses the method zerofree: base from A052382.
  Usage: n A043489
  Answer: a(n)"
  | n a b dj cj gj ej j r |
  n := self.
  n <= 1 ifTrue: [^r := 0].
  n <= 10 ifTrue: [^r := (n - 1) * 10].
  j := n invGeometricSum2: 9.
  b := j geometricSum2: 9.
  cj := 9 ** j.
  dj := (j + 1) * cj.
  gj := (cj - 1) / 8.
  ej := 10 ** j.
  a := n - b - 2.
  b := a \\ dj.
  r := (a // dj + 1) * ej * 10.
  [b >= cj] whileTrue:
          [a := b - cj.
          cj := cj // 9.
          dj := j * cj.
          b := a \\ dj.
          r := (a // dj + 1) * ej + r.
          gj := gj - cj.
          ej := ej // 10.
          j := j - 1].
  r := (b + gj zerofree: 10) + r.
  ^r
[by Hieronymus Fischer, May 28 2014]
------------------

A043489_inverse
  "Answers the index n such that A043489(n) = m, where m is the receiver. Uses A052382_inverse from A052382.
  Usage: n zerofree_inverse: b [b = 10 for this sequence]
  Answer: a(n)"
  | m p q s r m1 mr |
  m := self.
  m < 100 ifTrue: [^m // 10 + 1].
  p := q := 1.
  s := 0.
  [m // p \\ 10 = 0] whileFalse:
     [p := 10 * p.
     s := s + q.
     q := 9 * q].
  p > 1
     ifTrue:
     [r := m \\ p.
     p := 10 * p.
     m1 := m // p.
     (m1 \\ 10 = 1 and: [m1 > 10])
          ifTrue: [mr := m - r - 1]
          ifFalse: [mr := m - r - 10].
     ^mr A043489_inverse + r A052382_inverse - s + 1]
     ifFalse:
     [s := 1.
     p := 10.
     q := 1.
     [p < m] whileTrue:
          [s := (m // p) A052382_inverse * q + s.
          p := 10 * p.
          q := 9 * q].
     ^s]
[by Hieronymus Fischer, May 28 2014]

From Hieronymus Fischer, May 28 2014: (Start)
a(1 + Sum_{j=1..n} j*9^j) = 10*(10^n - 1).
a(2 + Sum_{j=1..n} j*9^j) = 10^(n+1) + (10^n - 1)/9 = (91*10^n - 1)/9.
a((9^(n+1) - 1)/8 + 1 + Sum_{j=1..n} j*9^j) = 10*(10^(n+1) - 1)/9, where Sum_{j=1..n} j*9^j = (1-(n+1)*9^n+n*9^(n+1))*9/64.
Iterative calculation:
With i := digit position of the '0' in a(n) counted from the right (starting with 0), j = number of contiguous '9' digits in a(n) counted from position 1, if i = 0, and counted from position 0, if i > 0 (0 if none)
a(n+1) = a(n) + 10 + (10^j - 1)/9, if i = 0.
a(n+1) = a(n) + 1 + (10^(j-1) - 1)/9, if i = j > 0.
a(n+1) = a(n) + 1 + (10^j - 1)/9, if i > j.
[see Prog section for an implementation in Smalltalk].
Direct calculation:
Set j := max( m | (Sum_{i=1..m} i*9^i) < n) and c(1) := n - 2 - Sum_{i=1..j} i*9^i. Define successively,
c(i+1) = c(i) mod ((j-i+2)*9^(j-i+1)) - 9^(j-i+1) while this value is >= 0, and set k := i for the last such index for which c(i) >= 0.
Then a(n) = A052382(c(k) mod ((j-k+2)*9^(j-k+1)) + (9^(j-k+1)-1)/8) + Sum_{i=1..k} ((floor(c(i)/((j-i+2)*9^(j-i+1))) + 1) * 10^(j-i+2)). [see Prog section for an implementation in Smalltalk].
Behavior for large n:
a(n) = O(n^(log(10)/log(9))/log(n)).
a(n) = O(n^1.047951651.../log(n)).
Inequalities:
a(n) < 2*(8n)^log_9(10)/(log_9(8n)*log_9(10)).
a(n) < (8n)^log_9(10)/(log_9(8n)*log_9(10)), for large n (n > 10^50).
a(n) > 0.9*(8n)^log_9(10)/(log_9(8n)*log_9(10)), for 2 < n < 10^50.
a(n) >= A011540(n), equality holds for n <= 10.
(End)

A043493 Numbers that contain a single 1.

Original entry on oeis.org

1, 10, 12, 13, 14, 15, 16, 17, 18, 19, 21, 31, 41, 51, 61, 71, 81, 91, 100, 102, 103, 104, 105, 106, 107, 108, 109, 120, 122, 123, 124, 125, 126, 127, 128, 129, 130, 132, 133, 134, 135, 136, 137, 138, 139, 140, 142, 143, 144, 145

Offset: 1

Clark Kimberling

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

A178550 is a subsequence.

Cf. A043489, A043497, A043501, A043505, A043509, A043513, A043517, A043521, A043525, A011531.

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

a(n) >> n^k where k = log(10)/log(9) = 1.04795.... - Charles R Greathouse IV, Jan 21 2025

A043509 Numbers having exactly one 5 in base 10.

Original entry on oeis.org

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

A043513 Numbers having one 6 in base 10.

Original entry on oeis.org

6, 16, 26, 36, 46, 56, 60, 61, 62, 63, 64, 65, 67, 68, 69, 76, 86, 96, 106, 116, 126, 136, 146, 156, 160, 161, 162, 163, 164, 165, 167, 168, 169, 176, 186, 196, 206, 216, 226, 236, 246, 256, 260, 261, 262, 263, 264, 265, 267, 268

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, A043509, A043517, A043521, A043525, A011536.

Select[Range[300],DigitCount[#,10,6]==1&] (* Harvey P. Dale, Aug 15 2011 *)

A043497 Numbers having one 2 in base 10.

Original entry on oeis.org

2, 12, 20, 21, 23, 24, 25, 26, 27, 28, 29, 32, 42, 52, 62, 72, 82, 92, 102, 112, 120, 121, 123, 124, 125, 126, 127, 128, 129, 132, 142, 152, 162, 172, 182, 192, 200, 201, 203, 204, 205, 206, 207, 208, 209, 210, 211, 213, 214, 215

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, A043501, A043505, A043509, A043513, A043517, A043521, A043525, A011532.

Select[Range[9000],DigitCount[#,10,2]==1&] (* Enrique Pérez Herrero, Nov 29 2013 *)

A043501 Numbers having one 3 in base 10.

Original entry on oeis.org

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

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, A043505, A043509, A043513, A043517, A043521, A043525, A011533.

Select[Range[9000],DigitCount[#,10,3]==1&] (* Enrique Pérez Herrero, Nov 29 2013 *)

A043505 Numbers having one 4 in base 10.

Original entry on oeis.org

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

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, A043509, A043513, A043517, A043521, A043525, A011534.

Select[Range[1000],DigitCount[#,10,4]==1&] (* Enrique Pérez Herrero, Nov 29 2013 *)

A043517 Numbers having one 7 in base 10.

Original entry on oeis.org

7, 17, 27, 37, 47, 57, 67, 70, 71, 72, 73, 74, 75, 76, 78, 79, 87, 97, 107, 117, 127, 137, 147, 157, 167, 170, 171, 172, 173, 174, 175, 176, 178, 179, 187, 197, 207, 217, 227, 237, 247, 257, 267, 270, 271, 272, 273, 274, 275, 276

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, A043509, A043513, A043521, A043525, A011537.

Select[Range[9000],DigitCount[#,10,7]==1&]

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

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, A043509, A043513, A043517, A043521.
Cf. A011539, A140502.
Subsequence of A011539.

Select[Range[300],DigitCount[#,10,9]==1&] (* Harvey P. Dale, Jan 19 2013 *)

def ok(n): return str(n).count('9') == 1
print(list(filter(ok, range(320)))) # Michael S. Branicky, Sep 19 2021

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

cp's OEIS Frontend

A011538 Numbers that contain an 8.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Python

Formula

A043489 Numbers having one 0 in base 10.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Smalltalk

Smalltalk

Smalltalk

Formula

A043493 Numbers that contain a single 1.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A043509 Numbers having exactly one 5 in base 10.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A043513 Numbers having one 6 in base 10.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A043497 Numbers having one 2 in base 10.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A043501 Numbers having one 3 in base 10.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A043505 Numbers having one 4 in base 10.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica