A171102 Pandigital numbers: numbers containing the digits 0-9. Version 2: each digit appears at least once.

1023456789, 1023456798, 1023456879, 1023456897, 1023456978, 1023456987, 1023457689, 1023457698, 1023457869, 1023457896, 1023457968, 1023457986, 1023458679, 1023458697, 1023458769, 1023458796, 1023458967, 1023458976

Offset: 1

Author

N. J. A. Sloane, Sep 25 2010

Keywords

• nonn
• base

Comments

This is the infinite version. See A050278 for the finite version.

The first 9*9!=3265920 terms of this sequence are permutations of the digits 0-9 with a(9*9!)=9876543210 (see Version 1, A050278). - Jeremy Gardiner, May 29 2010

Subsequence of A134336 and of A178403; A178401(a(n))>0. - Reinhard Zumkeller, May 27 2010

Smallest prime factors: A178775(n) = A020639(a(n)). - Reinhard Zumkeller, Jun 11 2010

A178788(a(n)) = 1, for n <= 9*9!, else A178788(a(n)) = 0. - Reinhard Zumkeller, Jun 30 2010 [corrected by Hieronymus Fischer, Feb 02 2013]

A230959(a(n)) = 0. - Reinhard Zumkeller, Nov 02 2013

The first term of the sequence absent in A050278 is a(3265921) = 10123456789. Also, the first prime is a(3306373) = 10123457689 = A050288(1). - Zak Seidov, Sep 23 2015

Almost all numbers are in this sequence, in the sense that it has asymptotic density equal to 1. Indeed, the fraction of n-digit numbers which don't have a given digit d is roughly 0.9^n (not exactly because the first digit is chosen among {1..9}) which tends to zero as n -> oo. - M. F. Hasler, Jan 05 2020

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..1000 . [From _Robert G. Wilson v_, May 30 2010]
• Eric Weisstein's World of Mathematics, Pandigital Number.
• Chai Wah Wu, Pandigital and penholodigital numbers, arXiv:2403.20304 [math.GM], 2024. See p. 1.

Crossrefs

Cf. A050278, A050288, A050289.

Cf. A051264, A051018, A051019, A051020.

Subsequence of A253172.

Programs

Mathematica

Take[ Select[ FromDigits@# & /@ Permutations[ Range[0, 9], {10}], # > 10^9 &], 20] (* Robert G. Wilson v, May 30 2010 *)

PARI

is_A171102(n)=9<#vecsort(Vecsmall(Str(n)),,8) /* assuming that n is a nonnegative integer. In PARI/GP V.2.4 - 2.9 this is faster than other possibilities involving Set(),Vec(),eval() or digits() */ \\ M. F. Hasler, Jan 10 2012, Sep 19 2017

PARI

A171102=A050278 /*** valid for n <= 9*9! ***/ \\ M. F. Hasler, Jan 10 2012

Formula

a(n) = 1011111111 + A178478(n) for n = 1,...,8!. - M. F. Hasler, Jan 10 2012

A171102(n) = A050278(n) for n <= 9*9!.