A043493 -id:A043493 - cp's OEIS frontend

A011531 Numbers that contain a digit 1 in their decimal representation.

1, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 31, 41, 51, 61, 71, 81, 91, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133

Offset: 1

N. J. A. Sloane

A121042(a(n)) = 1. - Reinhard Zumkeller, Jul 21 2006

See A043493 for numbers that contain a single digit '1'. A subsequence of numbers having a digit that divides all other digits, A263314. - M. F. Hasler, Jan 11 2016

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

Cf. A121041, A121022, A121023, A121024, A121025, A121026, A121027, A121028, A121029, A121030, A121031, A121032, A121033, A121034, A121035, A121036, A121037, A121038, A121039, A121040.

Cf. A052383 (complement), A062634 (a subsequence).

Filtered([1..140],n->1 in ListOfDigits(n)); # Muniru A Asiru, Feb 23 2019

a011531 n = a011531_list !! (n-1)
a011531_list = filter ((elem '1') . show) [0..]
-- Reinhard Zumkeller, Feb 05 2012

[n: n in [0..500] | 1 in Intseq(n) ]; // Vincenzo Librandi, Jan 11 2016

M:= 3: # to get all terms of up to M digits
B:= {1}: A:= {1}:
for i from 2 to M do
   B:= map(t -> seq(10*t+j,j=0..9),B) union
      {seq(10*x+1,x=2*10^(i-2)..10^(i-1)-1)}:
   A:= A union B;
od:
sort(convert(A,list)); # Robert Israel, Jan 10 2016
# second program:
A011531 := proc(n)
    if n = 1 then
        1;
    else
        for a from procname(n-1)+1 do
            if nops(convert(convert(a,base,10),set) intersect {1}) > 0 then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Jul 31 2016

Select[Range[600] - 1, DigitCount[#, 10, 1] > 0 &] (* Vincenzo Librandi, Jan 11 2016 *)

is_A011531(n)=setsearch(Set(digits(n)),1) \\ M. F. Hasler, Jan 10 2016

def aupto(nn): return [m for m in range(1, nn+1) if '1' in str(m)]
print(aupto(133)) # Michael S. Branicky, Jan 10 2021

(0 to 119).filter(.toString.indexOf('1') > -1) // _Alonso del Arte, Jan 12 2020

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

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)

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

A178550 Primes with exactly one digit 1.

Original entry on oeis.org

13, 17, 19, 31, 41, 61, 71, 103, 107, 109, 127, 137, 139, 149, 157, 163, 167, 173, 179, 193, 197, 199, 241, 251, 271, 281, 313, 317, 331, 401, 419, 421, 431, 461, 491, 521, 541, 571, 601, 613, 617, 619, 631, 641, 661, 691, 701, 719, 751, 761, 821, 881, 919, 941, 971, 991

Offset: 1

Lekraj Beedassy, May 29 2010

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Intersection of A043493 and A000040.

Cf. A038618, A178551, A178552, A178553, A178554, A178555, A178556, A178557, A178558.

Select[Prime[Range[200]],Count[IntegerDigits[#],1]==1 &] (* Stefano Spezia, Aug 29 2025 *)

from sympy import isprime
print([i for i in range(1000) if str(i).count('1') == 1 and isprime(i)]) # Daniel Starodubtsev, Mar 29 2020

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

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

A011531 Numbers that contain a digit 1 in their decimal representation.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Haskell

Magma

Maple

Mathematica

PARI

Python

Scala

Formula

A043489 Numbers having one 0 in base 10.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Smalltalk

Smalltalk

Smalltalk

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

A043521 Numbers having one 8 in base 10.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Python

A178550 Primes with exactly one digit 1.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Python

Formula

A043497 Numbers having one 2 in base 10.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A043501 Numbers having one 3 in base 10.

Views