A051020 -id:A051020 - cp's OEIS frontend

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

N. J. A. Sloane, Sep 25 2010

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

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.

Cf. A050278, A050288, A050289.
Cf. A051264, A051018, A051019, A051020.
Subsequence of A253172.

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

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

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

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

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

A051018 Numbers that are 2-persistent but not 3-persistent.

Original entry on oeis.org

1023456789, 1023456879, 1023457689, 1023457869, 1023458679, 1023458769, 1023465789, 1023465879, 1023467589, 1023467859, 1023468579, 1023468759, 1023475689, 1023475869, 1023476589, 1023476859, 1023478569, 1023478659, 1023485679, 1023485769, 1023486579, 1023486759

Offset: 1

Eric W. Weisstein

A number m is k-persistent iff all of {m, 2m,..., km} are pandigital (in the sense of A171102).

Hans Havermann, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Persistent Number

Cf. A171102 (pandigital), A204047 (smallest n-persistent), A051264 (1-persistent), A051019 (3-persistent), A051020 (4-persistent), A204096 (5-persistent), A204097 (6-persistent).

is_A051018(n,k=3)=10>#Set(Vec(Str(k*n))) & !while(k--,9<#Set(Vec(Str(k*n))) || return(0)) \\ M. F. Hasler, Jan 10 2012

Definition corrected by Franklin T. Adams-Watters, Jan 09 2012

A051019 Numbers that are 3-persistent but not 4-persistent.

Original entry on oeis.org

1052674893, 1052687493, 1052746893, 1052748693, 1052867493, 1052874693, 1053267489, 1053268749, 1053274869, 1053286749, 1053287469, 1065273489, 1065287349, 1067285493, 1067328549, 1068547293, 1068547329, 1068549273

Offset: 1

Eric W. Weisstein

A number n is k-persistent iff all of {n, 2n,..., kn} are pandigital (in the sense of A171102).

Hans Havermann, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Persistent Number

Cf. A171102 (pandigital), A204047 (smallest n-persistent), A051264 (1-persistent), A051018 (2-persistent), A051020 (4-persistent), A204096 (5-persistent), A204097 (6-persistent).

is_A051019(n)=for(i=1,4, 9<#Set(Vec(Str(i*n))) || return(i>3)) \\ M. F. Hasler, Jan 10 2012

Definition corrected by Franklin T. Adams-Watters, Jan 09 2012

A051264 Numbers that are 1-persistent but not 2-persistent.

Original entry on oeis.org

1023456798, 1023456897, 1023456978, 1023456987, 1023457698, 1023457896, 1023457968, 1023457986, 1023458697, 1023458796, 1023458967, 1023458976, 1023459678, 1023459687, 1023459768, 1023459786, 1023459867, 1023459876

Offset: 1

Eric W. Weisstein

A number n is k-persistent iff all of {n, 2n,..., kn} are pandigital (in the sense of A171102).

Hans Havermann, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Persistent Number

Cf. A171102 (1-persistent, i.e. pandigital numbers), A204047 (smallest n-persistent), A051018 (2-persistent), A051019 (3-persistent), A051020 (4-persistent), A204096 (5-persistent), A204097 (6-persistent).

A051264 = A171102 \ A051018. - M. F. Hasler, Jan 09 2012

Definition corrected by Franklin T. Adams-Watters, Jan 09 2012

A204047 Smallest number that is n-persistent but not (n+1)-persistent, i.e., k, 2k, ..., nk, but not (n+1)k, are pandigital in the sense of A171102; 0 if such a number does not exist.

Original entry on oeis.org

1023456798, 1023456789, 1052674893, 1053274689, 13047685942, 36492195078, 153846076923, 251793406487, 0, 1189658042735, 5128207435967, 3846154076923, 125583660720493, 125583660493072, 180106284973592, 201062849735918

Offset: 1

Hans Havermann, Jan 09 2012

a(9) is 0 because any 9-persistent number is also 10-persistent. Indeed, if n is pandigital, 10*n is pandigital as well.

In the same way, a(10m-1)=0 for all m>0 since if kn is pandigital for all k=1,...,10m-1, then mn is pandigital and so is 10mn. - M. F. Hasler, Jan 10 2012

			k=36492195078 is the smallest number such that k, 2k, 3k, 4k, 5k, and 6k, each contain all ten digits, but 7k=255445365546 contains only five of the ten, so a(6)= 36492195078.

Ross Honsberger, More Mathematical Morsels, Mathematical Association of America, 1991, pages 15-18.

Cf. A051264, A051018, A051019, A051020, A204096, A204097.

a(7)-a(16) from Giovanni Resta, Jan 10 2012

A204096 Numbers that are 5-persistent but not 6-persistent.

Original entry on oeis.org

13047685942, 14057869523, 20476859413, 21304768594, 35078196492, 35079219648, 35079648192, 35079649218, 35081964792, 35092196478, 35096478192, 35096479218, 35180796492, 35180964792, 35192079648, 35192096478, 35196478092, 35196479208

Offset: 1

Hans Havermann, Jan 10 2012

A number n is k-persistent iff all of {n, 2n,..., kn} are pandigital (in the sense of A171102).

Ross Honsberger, More Mathematical Morsels, Mathematical Association of America, 1991, pages 15-18.

Hans Havermann, Table of n, a(n) for n = 1..1000

Cf. A171102 (pandigital), A051264 (1-persistent), A051018 (2-persistent), A051019 (3-persistent), A051020 (4-persistent), A204097 (6-persistent), A204047 (smallest n-persistent).

A204097 Numbers that are 6-persistent but not 7-persistent.

Original entry on oeis.org

36492195078, 48602175913, 48613021759, 49021758613, 49130217586, 49219635078, 53829197460, 53829301746, 53928301746, 54601738293, 54601739283, 58829301746, 59288301746, 60174538293, 60174539283, 60174588293, 60174592883, 64820935179

Offset: 1

Hans Havermann, Jan 10 2012

A number n is k-persistent iff all of {n, 2n,..., kn} are pandigital (in the sense of A171102).

Ross Honsberger, More Mathematical Morsels, Mathematical Association of America, 1991, pages 15-18.

Hans Havermann, Table of n, a(n) for n = 1..1000

Cf. A171102 (pandigital), A051264 (1-persistent), A051018 (2-persistent), A051019 (3-persistent), A051020 (4-persistent), A204096 (5-persistent), A204047 (smallest n-persistent).

A180618 The next smallest pandigital multiple of the pandigital number A050278(n).

Original entry on oeis.org

2046913578, 10234567980, 2046913758, 10234568970, 10234569780, 10234569870, 2046915378, 10234576980, 2046915738, 10234578960, 10234579680, 10234579860, 2046917358, 10234586970, 2046917538, 10234587960, 10234589670, 10234589760

Offset: 1

Lekraj Beedassy, Sep 12 2010

The smallest number of the form k*A050278(n), k>=2, which is in A171102.

Cf. A051018, A051019, A051020

Keyword:base and 2 different pandigital definitions introduced by R. J. Mathar, Oct 06 2010

cp's OEIS Frontend

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

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A051018 Numbers that are 2-persistent but not 3-persistent.

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A051019 Numbers that are 3-persistent but not 4-persistent.

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A051264 Numbers that are 1-persistent but not 2-persistent.

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A204047 Smallest number that is n-persistent but not (n+1)-persistent, i.e., k, 2k, ..., nk, but not (n+1)k, are pandigital in the sense of A171102; 0 if such a number does not exist.

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A204096 Numbers that are 5-persistent but not 6-persistent.

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A204097 Numbers that are 6-persistent but not 7-persistent.

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A180618 The next smallest pandigital multiple of the pandigital number A050278(n).

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