A043525 -id:A043525 - cp's OEIS frontend

A011539 "9ish numbers": decimal representation contains at least one nine.

9, 19, 29, 39, 49, 59, 69, 79, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 109, 119, 129, 139, 149, 159, 169, 179, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 209, 219, 229, 239, 249, 259, 269, 279, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298

Offset: 1

N. J. A. Sloane

The 9ish numbers are closed under lunar multiplication. The lunar primes (A087097) are a subset.
Almost all numbers are 9ish, in the sense that the asymptotic density of this set is 1: Among the 9*10^(n-1) n-digit numbers, only a fraction of 0.8*0.9^(n-1) doesn't have a digit 9, and this fraction tends to zero (< 1/10^k for n > 22k-3). This explains the formula a(n) ~ n. - M. F. Hasler, Nov 19 2018
A 9ish number is a number whose largest decimal digit is 9. - Stefano Spezia, Nov 16 2023

			E.g. 9, 19, 69, 90, 96, 99 and 1234567890 are all 9ish.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
D. Applegate, C program for lunar arithmetic and number theory [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
D. Applegate, M. LeBrun, and N. J. A. Sloane, Dismal Arithmetic, arXiv:1107.1130 [math.NT], 2011. [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
Index entries for sequences related to dismal (or lunar) arithmetic
Index entries for 10-automatic sequences.

Cf. A088924 (number of n-digit terms).
Cf. A087062 (lunar product), A087097 (lunar primes).
A102683 (number of digits 9 in n); fixed points > 8 of A068505.
Cf. Numbers with at least one digit b-1 in base b : A074940 (b=3), A337250 (b=4), A337572 (b=5), A333656 (b=6), A337141 (b=7), A337239 (b=8), A338090 (b=9), this sequence (b=10), A095778 (b=11).
Cf. Numbers with no digit b-1 in base b: A005836 (b=3), A023717 (b=4), A020654 (b=5), A037465 (b=6), A020657 (b=7), A037474 (b=8), A037477 (b=9), A007095 (b=10), A171397 (b=11).
Supersequence of A043525.

Filtered([1..300],n->9 in ListOfDigits(n)); # Muniru A Asiru, Feb 25 2019

a011539 n = a011539_list !! (n-1)
a011539_list = filter ((> 0) . a102683) [1..]  -- Reinhard Zumkeller, Dec 29 2011

seq(`if`(numboccur(9, convert(n, base, 10))>0, n, NULL), n=0..100); # François Marques, Oct 12 2020

Select[ Range[ 0, 100 ], (Count[ IntegerDigits[ #, 10 ], 9 ]>0)& ] (* François Marques, Oct 12 2020 *)
Select[Range[300],DigitCount[#,10,9]>0&] (* Harvey P. Dale, Mar 04 2023 *)

is(n)=n=vecsort(digits(n));n[#n]==9 \\ Charles R Greathouse IV, May 15 2013

select( is_A011539(n)=vecmax(digits(n))==9, [1..300]) \\ M. F. Hasler, Nov 16 2018

def ok(n): return '9' in str(n)
print(list(filter(ok, range(299)))) # Michael S. Branicky, Sep 19 2021

def A011539(n):
    def f(x):
        l = (s:=str(x)).find('9')
        if l >= 0: s = s[:l]+'8'*(len(s)-l)
        return n+int(s,9)
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Dec 04 2024

Complement of A007095. A102683(a(n)) > 0 (defines this sequence). A068505(a(n)) = a(n): fixed points of A068505 are the terms of this sequence and the numbers < 9. - Reinhard Zumkeller, Dec 29 2011, edited by M. F. Hasler, Nov 16 2018

a(n) ~ n. - Charles R Greathouse IV, May 15 2013

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 *)

A043521 Numbers having one 8 in base 10.

Original entry on oeis.org

8, 18, 28, 38, 48, 58, 68, 78, 80, 81, 82, 83, 84, 85, 86, 87, 89, 98, 108, 118, 128, 138, 148, 158, 168, 178, 180, 181, 182, 183, 184, 185, 186, 187, 189, 198, 208, 218, 228, 238, 248, 258, 268, 278, 280, 281, 282, 283, 284, 285, 286, 287, 289, 298, 308, 318

Offset: 1

Clark Kimberling

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

Subsequence of A011538.

Cf. A043489, A043493, A043497, A043501, A043505, A043509, A043513, A043517, A043525, A011538.

Select[Range[300],DigitCount[#,10,8]==1&] (* Harvey P. Dale, Jan 06 2012 *)

is(n)=my(d=digits(n)); sum(i=1,#d, d[i]==8)==1 \\ Charles R Greathouse IV, Feb 12 2017

def ok(n): return str(n).count('8') == 1
print(list(filter(ok, range(320)))) # Michael S. Branicky, Aug 18 2021

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&]

cp's OEIS Frontend

A011539 "9ish numbers": decimal representation contains at least one nine.

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A043489 Numbers having one 0 in base 10.

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A043493 Numbers that contain a single 1.

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A043509 Numbers having exactly one 5 in base 10.

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A043513 Numbers having one 6 in base 10.

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A043521 Numbers having one 8 in base 10.

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A043497 Numbers having one 2 in base 10.

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A043501 Numbers having one 3 in base 10.

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