A230959 -id:A230959 - 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!.

A050289 Zeroless pandigital numbers: numbers containing the digits 1-9 (each appearing at least once) and no 0's.

Original entry on oeis.org

123456789, 123456798, 123456879, 123456897, 123456978, 123456987, 123457689, 123457698, 123457869, 123457896, 123457968, 123457986, 123458679, 123458697, 123458769, 123458796, 123458967, 123458976, 123459678, 123459687, 123459768, 123459786, 123459867, 123459876, 123465789

Offset: 1

Eric W. Weisstein

The first 9! = 362880 terms of this sequence are permutations of the digits 1-9 with a(9!) = 987654321. - Jeremy Gardiner, May 28 2010
First differences are given in A209280 (for the first 9! terms) or in A219664 (for at least as much initial terms). - M. F. Hasler, Mar 03 2013
A230959(a(n)) = 0. - Reinhard Zumkeller, Nov 02 2013
After the first 9! terms, 8! + 7! = 9*7! of the initial terms are repeated with a leading '1' prefixed, cf. formula. However, a(9!+8!+7!) = 1219...3 is followed by 122...9 and permutations of the last 7 digits, before 12314..9. - M. F. Hasler, Jan 08 2020, corrected Aug 11 2022 thanks to a remark from Michael S. Branicky

H. Fripertinger, Operate on "9" to display zeroless pandigitals
James Grime and Brady Haran, Why 381,654,729 is awesome, Numberphile video (2013).
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. A050290, A133360, A171102.

apply( {A050289(n)=if(n<=7!*81, fromdigits(Vec(numtoperm(9,n-1)))+(n-1)\9!*10^9, "not yet implemented")}, [1..25]) \\ M. F. Hasler, Jan 07 2020, corrected Aug 11 2022

from itertools import count, islice, permutations, product
def c(t): return len(set(t)) == 9
def t2i(t): return int("".join(map(str, t)))
def agen():
    yield from (t2i(p) for p in permutations(range(1, 10)))
    for d in count(10):
        yield from (t2i(p) for p in product(range(1, 10), repeat=d) if c(p))
print(list(islice(agen(), 25))) # Michael S. Branicky, May 30 2022, updated Aug 05 2022

a(n + 9!) = a(n) + 10^9 for 1 <= n <= 8! + 7!. - M. F. Hasler, Jan 08 2020, corrected Aug 11 2022

Name clarified by Michael S. Branicky, Aug 05 2022

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

A257001 Numbers such that the largest missing digit is a divisor.

Original entry on oeis.org

18, 27, 36, 45, 54, 63, 72, 81, 96, 98, 108, 117, 126, 135, 144, 153, 162, 171, 180, 189, 192, 207, 216, 225, 234, 243, 252, 261, 270, 288, 296, 306, 315, 324, 333, 342, 351, 360, 378, 387, 392, 405, 414, 423, 432, 441, 450, 468, 477, 486, 496, 504, 513, 522

Offset: 1

Eric Angelini and Reinhard Zumkeller, Apr 14 2015

a(n) mod A000030(A230959(a(n))) = 0.
Pandigital numbers are not terms: A171102, A050278.
The original definition used the phrase "largest absent digit".

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Éric Angelini, Absent digit, SeqFan list, Apr 14 2015.

Cf. A000030, A230959, A050278, A116667, A171102.

import Data.List ((\\)); import Data.Char (digitToInt)
a257001 n = a257001_list !! (n-1)
a257001_list = filter f [1..] where
   f x = h > 0 && mod x h == 0 where h = a000030 $ a230959 x

f[x_]:=Union[Sort[IntegerDigits[x]]];
d={1,2,3,4,5,6,7,8,9};
Select[Range[525],And[f[#]!=d,Length[f[#]]<10,IntegerQ[#/Max[Complement[d,f[#]]]]]&] (* Ivan N. Ianakiev, Apr 14 2015 *)

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

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A050289 Zeroless pandigital numbers: numbers containing the digits 1-9 (each appearing at least once) and no 0's.

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A227362 Distinct digits of n arranged in decreasing order.

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A257001 Numbers such that the largest missing digit is a divisor.

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