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

Author

N. J. A. Sloane

Keywords

• nonn
• base
• fini

Comments

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

Links

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

Crossrefs

Subsequence of A043096.

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

Programs

Haskell

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

Mathematica

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

PARI

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

PARI

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

PARI

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

Python

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

Python

# 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

Scala

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

Formula

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

Extensions

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