A109303 -id:A109303 - cp's OEIS frontend

A010784 Numbers with distinct decimal digits.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 102, 103, 104, 105, 106, 107, 108, 109, 120

Offset: 1

N. J. A. Sloane

More than the usual number of terms are displayed in order to show the difference from some closely related sequences.
Also: a(1) = 0; a(n) = Min{x integer | x > a(n-1) and all digits to base 10 are distinct}.
This sequence is finite: a(8877691) = 9876543210 is the last term; a(8877690) = 9876543201. The largest gap between two consecutive terms before a(249999) = 2409653 is 104691, as a(175289) = 1098765, a(175290) = 1203456. - Reinhard Zumkeller, Jun 23 2001
Complement of A109303. - David Wasserman, May 21 2008
For the analogs in other bases b, search for "xenodromes."  A001339(b-1) is the number of base b xenodromes for b >= 2. - Rick L. Shepherd, Feb 16 2013
A073531 gives the number of positive n-digit numbers in this sequence. Note that it does not count 0. - T. D. Noe, Jul 09 2013
Can be seen as irregular table whose n-th row holds the n-digit terms; length of row n is then A073531(n) = 9*9!/(10-n)! except for n = 1 where we have 10 terms, unless 0 is considered to belong to a row 0. - M. F. Hasler, Dec 10 2018

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Digit

Subsequence of A043096.

Cf. A109303, A029740 (odds), A029741 (evens), A029743 (primes), A001339.

a010784 n = a010784_list !! (n-1)
a010784_list = filter ((== 1) . a178788) [1..]
-- Reinhard Zumkeller, Sep 29 2011

Select[Range[0,100], Max[DigitCount[#]] == 1 &] (* Harvey P. Dale, Apr 04 2013 *)

is(n)=my(v=vecsort(digits(n)));v==vecsort(v,,8) \\ Charles R Greathouse IV, Sep 17 2012

select( is(n)=!n||#Set(digits(n))==logint(n,10)+1, [0..120]) \\ M. F. Hasler, Dec 10 2018

apply( A010784_row(n,L=List(if(n>1,[])))={forvec(d=vector(n,i,[0,9]),forperm(d,p,p[1]&&listput(L,fromdigits(Vec(p)))),2);Set(L)}, [1..2]) \\ A010784_row(n) returns all terms with n digits. - M. F. Hasler, Dec 10 2018

A010784_list = [n for n in range(10**6) if len(set(str(n))) == len(str(n))] # Chai Wah Wu, Oct 13 2019

# alternate for generating full sequence
from itertools import permutations
afull = [0] + [int("".join(p)) for d in range(1, 11) for p in permutations("0123456789", d) if p[0] != "0"]
print(afull[:100]) # Michael S. Branicky, Aug 04 2022

def hasDistinctDigits(n: Int): Boolean = {
  val numerStr = n.toString
  val digitSet = numerStr.split("").toSet
  numerStr.length == digitSet.size
}
(0 to 99).filter(hasDistinctDigits) // Alonso del Arte, Jan 09 2020

A178788(a(n)) = 1; A178787(a(n)) = n; A043537(a(n)) = A055642(a(n)). - Reinhard Zumkeller, Jun 30 2010

A107846(a(n)) = 0. - Reinhard Zumkeller, Jul 09 2013

Offset changed to 1 and first comment adjusted by Reinhard Zumkeller, Jun 14 2010

A178788 Characteristic function of numbers having distinct digits in their decimal representation.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1

Offset: 0

Reinhard Zumkeller, Jun 30 2010

a(A010784(n)) = 1; a(A109303(n)) = 0;
first differences of A178787.
a(n) <= A196368(n).
a(n) = 0 for n > 9*9!. - Hieronymus Fischer, Feb 02 2013

R. Zumkeller, Table of n, a(n) for n = 0..10000
Michael De Vlieger, Plot a(n) at (x,y) = (n mod 1000, -floor(n/1000)), n = 1..10^6, where black represents a(n) = 1 and white represents a(n) = 0.
Eric Weisstein's World of Mathematics, Digit
Index entries for characteristic functions

import Data.List (nub)
a178788 n = fromEnum $ nub (show n) == show n
-- Reinhard Zumkeller, Sep 25 2011

lst = {}; Do[If[Select[Tally[IntegerDigits[n]][[All, 2]], # > 1 &] == {}, AppendTo[lst, 1], AppendTo[lst, 0]], {n, 0, 104}]; lst (* Arkadiusz Wesolowski, May 10 2012 *)

a(n) = 0^(A055642(n)-A043537(n)).

A227362 Distinct digits of n arranged in decreasing order.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 21, 31, 41, 51, 61, 71, 81, 91, 20, 21, 2, 32, 42, 52, 62, 72, 82, 92, 30, 31, 32, 3, 43, 53, 63, 73, 83, 93, 40, 41, 42, 43, 4, 54, 64, 74, 84, 94, 50, 51, 52, 53, 54, 5, 65, 75, 85, 95, 60, 61, 62, 63, 64, 65, 6, 76, 86

Offset: 0

Reinhard Zumkeller, Jul 09 2013

a(n) <= 9876543210; a(a(n)) = a(n);
A055642(a(n)) <= 10;
A055642(a(n)) <= A055642(n), A055642(a(n)) = A055642(n) iff A178788(n) = 1;
a(A109303(n)) < A109303(n); a(A009995(n)) = A009995(n); a(A071589(n)) > A071589(n);
a(n) = A151949(n) + A180410(n).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A004186, A230959.

import Data.List (nub, sort)
a227362 = read . reverse . sort . nub . show :: Integer -> Integer

a:= n-> parse(cat(sort([{convert(n, base, 10)[]}[]], `>`)[])):
seq(a(n), n=0..68);  # Alois P. Heinz, Sep 21 2022

f[n_] := FromDigits[Reverse@ Union@ IntegerDigits@ n]; f /@ Range[0, 68] (* Michael De Vlieger, Apr 16 2015, corrected by Robert G. Wilson v *)

a(n) = {if (n == 0, d = [0], d = digits(n)); eval(subst(Pol(vecsort(d,,12)), x, 10));} \\ Michel Marcus, Apr 16 2015

a(n)=fromdigits(vecsort(digits(n),,12)) \\ Charles R Greathouse IV, Apr 16 2015

def A227362(n): return int(''.join(sorted(set(str(n)),reverse=True))) # Chai Wah Wu, Nov 23 2022

A137564 a(n) is the number formed by removing from n all duplicate digits except the leftmost copy of each.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 2, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 3, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 4, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 5, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 6, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 7, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 8, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 9, 10, 10, 102, 103, 104, 105, 106, 107, 108, 109, 10, 1, 12

Offset: 0

Rick L. Shepherd, Jan 25 2008

Differs from A106612: a(100) = 10, A106612(100) = 100.
Differs from A337864: a(101) = 10, A337864(101) = 101.
a(n)=n iff n is a term of A010784. a(n)A109303.
A010784 is the sequence of distinct terms in this sequence, thus 9876543210 is the largest term here also, as no digit occurs more than once in any given term. Each term except 0 appears infinitely often in this sequence. - Rick L. Shepherd, Oct 03 2020

			a(100)=10 as a (second) 0 digit is dropped. a(1211323171)=1237.
a(10...1) = 10 for any number of 0's and/or 1's in any order replacing the "..." in the term's index. - _Rick L. Shepherd_, Oct 03 2020

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000 (corrected by Andrew Howroyd at the suggestion of Rodolfo Kurchan and Omar E. Pol, Oct 04 2020)

Cf. A106612, A010784 (fixed points), A109303 (non-fixed).

Cf. A043529 (equivalent in binary, except at n=0), A337864.

Table[FromDigits@ DeleteDuplicates@ IntegerDigits@ n, {n, 74}] (* Michael De Vlieger, Jun 01 2016 *)

a(n)={my(d=digits(n)); fromdigits(vecextract(d, vecsort(vecsort(d,,9))))} \\ Andrew Howroyd, Oct 04 2020

sub a {my($n)=@; my @seen; $n =~ s{.}{!$seen[$&]++ && $&}eg; $n} # _Kevin Ryde, Oct 04 2020

def a(n):
    seen, out, s = set(), "", str(n)
    for d in s:
        if d not in seen: out += d; seen.add(d)
    return int(out)
print([a(n) for n in range(113)]) # Michael S. Branicky, Jul 23 2022

A178787 Number of numbers <= n having distinct digits in their decimal representation, cf. A010784.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 61, 62, 63, 64, 65, 66

Offset: 0

Reinhard Zumkeller, Jun 30 2010

a(m)<=a(n) for m=9876543210;
a(A010784(n)) = n; a(A109303(n)) = a(A109303(n)-1);
partial sums of A178788.

			a(12) = #{0,1,2,3,4,5,6,7,8,9,10,12} = 12;
a(24) = a(12) + #{13,14,15,16,17,18,19,20,21,23,24} = 23.

R. Zumkeller, Table of n, a(n) for n = 0..25000
Eric Weisstein's World of Mathematics, Digit

a178787 n = a178787_list !! n
a178787_list = scanl1 (+) a178788_list
-- Reinhard Zumkeller, Mar 15 2014

f[n_] := Length[ Select[ Range[0, n], Max[ DigitCount[#]] == 1 &]]; Array[f, 70, 0] (* Robert G. Wilson v, Dec 13 2016 after Harvey P. Dale in A010784 *)

A084050 Numbers n such that at least one permutation of the digits of n yields a palindrome.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 20, 22, 30, 33, 40, 44, 50, 55, 60, 66, 70, 77, 80, 88, 90, 99, 100, 101, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 121, 122, 131, 133, 141, 144, 151, 155, 161, 166, 171, 177, 181, 188, 191, 199, 200, 202, 211, 212, 220, 221

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 26 2003

Union of A037124 and numbers with at most one decimal digit occurring an odd number of times. a(281)=1001 is the first term greater than 90 not also a term of A044959. The terms greater than 90 are a subsequence of A109303. - Rick L. Shepherd, Jun 24 2005

Cf. A044959, A084051.

Cf. A037124 (numbers with only one nonzero digit), A109303 (numbers with at least one duplicate digit).

Corrected by Rick L. Shepherd, Jun 24 2005

A107846 Number of duplicate digits of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0

Offset: 0

Rick L. Shepherd, May 24 2005

a(A010784(n)) = 0; a(A109303(n)) > 0. - Reinhard Zumkeller, Jul 09 2013

			a(11) = 1 because 11 has two total decimal digits but only one distinct digit (1) and 2-1=1.
Similarly, a(3653135) = 7 (total digits) - 4 (distinct digits: 1,3,5,6) = 3 (There are three duplicate digits here, namely, 3, 3 and 5).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A055642 (Total decimal digits of n), A043537 (Distinct decimal digits of n).

import Data.List (sort, group)
a107846 = length . concatMap tail . group . sort . show :: Integer -> Int
-- Reinhard Zumkeller, Jul 09 2013

Table[Total[Select[DigitCount[n]-1,#>0&]],{n,0,120}] (* Harvey P. Dale, Jul 31 2013 *)

def a(n): return len(s:=str(n)) - len(set(s))
print([a(n) for n in range(105)]) # Michael S. Branicky, Jan 09 2023

a(n) = A055642(n) - A043537(n).

A353181 Numbers in which more than half of the digits are the same.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 100, 101, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 121, 122, 131, 133, 141, 144, 151, 155, 161, 166, 171, 177, 181, 188, 191, 199, 200, 202, 211, 212, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 232

Offset: 1

Zhining Yang, Apr 29 2022

Same as A044959 for terms <= 1000, and then differing at A044959(272) = 1001 which is not a term here (the next instead a(272) = 1011).

			1211 is a term: it has 4 digits, 3 of which are 1's, and 3/4 > 1/2.
1212 is not a term: it has 4 digits, no one of which appears more than twice.

Zhining Yang, Table of n, a(n) for n = 1..30000

Cf. A109303, A044959.

Select[Range[300], Max[DigitCount[#]] > IntegerLength[#]/2 &]

def ok(k):
    t=str(k)
    return(max(t.count(str(i)) for i in range(10))>(len(t)//2))
print([n for n in range(1, 201) if ok(n)])

A073064 Primes with non-distinct digits.

Original entry on oeis.org

11, 101, 113, 131, 151, 181, 191, 199, 211, 223, 227, 229, 233, 277, 311, 313, 331, 337, 353, 373, 383, 433, 443, 449, 499, 557, 577, 599, 661, 677, 727, 733, 757, 773, 787, 797, 811, 877, 881, 883, 887, 911, 919, 929, 977, 991, 997, 1009, 1013, 1019

Offset: 1

Zak Seidov, Aug 24 2002

A000040 INTERSECT A109303. - R. J. Mathar, May 01 2008
Comment from N. J. A. Sloane, Jan 22 2023 (Start)
A "nontrivial permutation" means any one of the m!-1 elements of S_m apart from the identity permutation.
This sequence consists of those primes that are fixed under at least one nontrivial permutation of its digits.
A prime p is in the sequence iff its decimal expansion p = d_1 d_2 ... d_m is such that there is a non-identity permutation pi in S_m with the property that p = d_pi(1) d_pi(2) ... d_pi(m). (End)

			a(1)=11 because 11 is the first prime not all digits of which are distinct; a(2)=101 because 101 is the second prime not all digits of which are distinct.

Cf. A000040, A030291, A073069, A109303.

A055642 := proc(n) max(ilog10(n)+1,1) ; end: A043537 := proc(n) nops(convert(convert(n,base,10),set)) ; end: isA109303 := proc(n) RETURN( A055642(n) > A043537(n) ) ; end: isA073064 := proc(n) RETURN(isprime(n) and isA109303(n) ) ; end: for n from 1 to 1019 do if isA073064(n) then printf("%d,",n) ; fi ; od: # R. J. Mathar, May 01 2008

ta=IntegerDigits[Prime[Range[1000]]]; ta2=Table[Length[ta[[i]]]>Length[Union[ta[[i]]]], {i, 1000}]; Prime[Flatten[Position[ta2, True]]]

A377948 Numbers that have at least 1 repeated decimal digit and whose decimal digits are nondecreasing as place value decreases.

Original entry on oeis.org

11, 22, 33, 44, 55, 66, 77, 88, 99, 111, 112, 113, 114, 115, 116, 117, 118, 119, 122, 133, 144, 155, 166, 177, 188, 199, 222, 223, 224, 225, 226, 227, 228, 229, 233, 244, 255, 266, 277, 288, 299, 333, 334, 335, 336, 337, 338, 339, 344, 355, 366, 377, 388, 399

Offset: 1

Michael De Vlieger, Nov 14 2024

Intersection of A009994 and A109303.

Does not intersect either A009993 or A009995.

Michael De Vlieger, Table of n, a(n) for n = 1..10938 (includes all numbers up to 7 digits long)

Cf. A009993, A009994, A009995, A109303, A178788.

Select[Range[10^6], And[CountDistinct[#] != Length[#], AllTrue[Differences[#], # >= 0 &]] &[IntegerDigits[#]] &]
(* More efficient program: *)
b = 10; mm = b - 1; nn = 14;
s = Table[Map[Position[#, 1][[All, 1]] &,
  Permutations@ Join[ConstantArray[1, r], ConstantArray[0, mm - r] ] ],
    {r, Min[mm, nn]}];
Union@ Flatten@ Table[
  w = Apply[Join, Permutations /@ IntegerPartitions[n, Min[mm, n - 1] ] ];
  Reap[Do[
    Sow[Table[FromDigits[Flatten@
      MapIndexed[ConstantArray[m[[First[#2] ]], #1] &,
      w[[i]]], b], {m, s[[Length[w[[i]]] ]]} ] ],
    {i, Length[w]}] ][[-1, 1]], {n, 2, nn}]

A178788(a(n)) = 0.

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A010784 Numbers with distinct decimal digits.

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A178788 Characteristic function of numbers having distinct digits in their decimal representation.

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A227362 Distinct digits of n arranged in decreasing order.

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A137564 a(n) is the number formed by removing from n all duplicate digits except the leftmost copy of each.

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A178787 Number of numbers <= n having distinct digits in their decimal representation, cf. A010784.

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A084050 Numbers n such that at least one permutation of the digits of n yields a palindrome.

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A107846 Number of duplicate digits of n.

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