A171102 -id:A171102 - cp's OEIS frontend

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.

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

A180489 Smallest pandigital number (A171102) divisible by the n-th prime A000040(n).

Original entry on oeis.org

1023456798, 1023456789, 1023467895, 1023456798, 1024375869, 1023456798, 1023457698, 1023458769, 1023475689, 1023468957, 1023458769, 1023654987, 1023458769, 1023469875, 1023467958, 1023459786, 1023457896, 1023458976

Offset: 1

Lekraj Beedassy, Sep 08 2010

Digits may appear more than once in the multiple, resulting in 11-or-more-digit values of a(n). The first entry for which that happens is a(10545), because the smallest multiple of the 10545th prime 111119 that contains all the digits 0-9 is 92373 * 111119 = 10264395387, and all smaller primes have 10-digit pandigital multiples. - David J. Seal, Sep 18 2017

			a(1) is the smallest pandigital number divisible by prime(1) = 2, which is 1023456798. - _David J. Seal_, Sep 18 2017

Ray Chandler, Table of n, a(n) for n=1..10000

Cf. A050278, A061604, A171102, A274328.

With[{s = Select[FromDigits@ # & /@ Permutations[Range[0, 9], {10}], # > 10^9 &]}, Table[SelectFirst[s, Divisible[#, Prime@ n] &], {n, 18}]] (* Michael De Vlieger, Sep 18 2017, after Robert G. Wilson v at A171102 *)

A175845 Pandigital numbers (A171102) of the form 9*prime.

Original entry on oeis.org

1023456879, 1023457689, 1023458697, 1023465789, 1023465879, 1023468759, 1023476589, 1023476859, 1023495867, 1023546789, 1023546987, 1023547869, 1023548769, 1023564879, 1023564987, 1023568479, 1023574689, 1023574869

Offset: 1

Zak Seidov, Sep 25 2010

Or pandigital numbers (A171102) that are 3-almost primes (product of three primes with repetition).
There are 160573 such integers.
Numbers of integers with first digit n=1..9 are:
15996,20815,15331,20042,19959,19723,14819,19422,14466.

Cf. A171102.

A276510 Numbers k such that the sum of all the different permutations of the digits of k (A045876(k)) is a pandigital number (a term of A171102).

Original entry on oeis.org

10234567, 10234576, 10234579, 10234597, 10234657, 10234675, 10234678, 10234687, 10234756, 10234759, 10234765, 10234768, 10234786, 10234795, 10234867, 10234876, 10234957, 10234975, 10235467, 10235476, 10235479, 10235497, 10235647, 10235674, 10235746, 10235749

Offset: 1

Altug Alkan, Sep 06 2016

			10234759 is a term because A045876(10234759) = 1567999984320, which contains every digit from 0 to 9.

Cf. A045876, A171102.

A047726(n) = n=eval(Vec(Str(n))); (#n)!/prod(i=0, 9, sum(j=1, #n, n[j]==i)!);
A055642(n) = #Str(n);
A007953(n) = sumdigits(n);
A045876(n) = ((10^A055642(n)-1)/9)*(A047726(n)*A007953(n)/A055642(n));
isA171102(n) = 9<#vecsort(Vecsmall(Str(n)), , 8);
is(n) = isA171102(A045876(n));

A292569 Least pandigital number which sums up with the n-th pandigital number A171102(n) to another pandigital number.

Original entry on oeis.org

1023456789, 1023456789, 1023456879, 1023456978, 1023456879, 1023456798, 1023457689, 1023457689, 1023457869, 1023459687, 1023457869, 1023459867, 1023458679, 1023459678, 1023458769, 1023456987, 1023459768, 1023456897, 1023458679, 1023457698, 1023458769

Offset: 1

M. F. Hasler, Sep 19 2017

The first 9*9! pandigital numbers (having each digit 0-9 exactly once) are listed in A050278, which is extended to the infinite sequence A171102 of pandigital numbers having each digit 0-9 at least once.

For all n, a(n) is well defined, because to any pandigital number N = A171102(n) we can add the number M(N) = 123456789*10^k with k = # digits of N, which is pandigital (in the above extended sense) as well as is the sum N + M(N) (equal to the concatenation of 123456789 and N). In practice, there are much smaller solutions. We conjecture that there is always a 10-digit solution a(n) < 10^10.

			The smallest pandigital number A171102(1) = A050278(1) = 1023456789, added to itself, yields again a pandigital number, 2046913578. Therefore, a(1) = A171102(1) = 1023456789.
Similarly, A171102(1) = 1023456789, added to the second pandigital number A171102(2) = 1023456798, yields the pandigital number 2046913587. Therefore also a(2) = A171102(1) = 1023456789.
Considering the third pandigital number A171102(3) = 1023456879, we have to add itself in order to get a pandigital number, 2046913758. (Adding A171102(1) or A171102(2) yields 2046913668 and 2046913677, respectively, which are not pandigital.) Therefore a(3) = A171102(3) = 1023456879.

Cf. A292570 (index of a(n) within A171102), A171102, A050278.

a(n)={n=A171102(n);for(k=1,9e9,#Set(digits(n+A171102(k)))>9&&return(A171102(k)))} \\ For illustrational purpose only; not optimized for efficiency.

a(n) = A171102(A292570(n)).

a(n) = min { N in A171102 | N + A171102(n) in A171102 }.

A292570 Least k > 0 such that A171102(n) + A171102(k) is again a term of A171102, the pandigital numbers (having each digit from '0' to '9' at least once).

Original entry on oeis.org

1, 1, 3, 5, 3, 2, 7, 7, 9, 20, 9, 23, 13, 19, 15, 6, 21, 4, 13, 8, 15, 17, 11, 14, 25, 25, 27, 29, 27, 26, 31, 31, 33, 78, 33, 76, 37, 43, 39, 92, 45, 95, 37, 32, 39, 86, 35, 89, 49, 49, 51, 98, 51, 101, 55, 55, 57, 18, 57, 16, 61, 104, 63, 24, 107, 22, 61, 115, 63, 10, 117, 12, 73, 97, 75, 30, 99, 28, 79, 103, 81, 116, 105, 119, 85, 44, 87, 102, 47, 100, 109, 38, 111, 113, 41, 110, 73, 50, 75, 77, 53, 74, 79, 56, 81, 62, 59, 65

Offset: 1

M. F. Hasler, Sep 19 2017

The first 9*9! pandigital numbers (having each digit 0-9 exactly once) are listed in A050278, which is extended to the infinite sequence A171102 of pandigital numbers having each digit 0-9 at least once.

For all n, a(n) is well defined, because to any pandigital number N = A171102(n) we can add the number M(N) = 123456789*10^k with k = # digits of N, which is pandigital (in the above extended sense) as well as is the sum N + M(N). In practice, there are much smaller solutions. We conjecture that there is always a 10-digit solution a(n) < 10^10.

			The smallest pandigital number A171102(1) = A050278(1) = 1023456789, added to itself, yields again a pandigital number, 2046913578. Therefore, a(1) = 1.
Similarly, A171102(1) = 1023456789 added to the second pandigital number A171102(2) = 1023456798, yields the pandigital number 2046913587. Therefore also a(2) = 1.
Considering the third pandigital number A171102(3) = 1023456879, we have to add itself in order to get a pandigital number, 2046913758. (Adding A171102(1) or A171102(2) yields 2046913668 and 2046913677, respectively, which are not pandigital.) Therefore a(3) = 3.

Cf. A292569 (the actual pandigital number to be added), A171102, A050278.

a(n)={n=A171102(n);for(k=1,oo,#Set(digits(n+A171102(k)))>9&&return(k))} \\ For illustrational purpose ; not optimized for efficiency.

a(n) = min { k in IN | A171102(k) + A171102(n) in A171102 }.

A338287 Decimal expansion of the sum of reciprocals of the numbers that are not pandigital numbers (version 2, A171102).

Original entry on oeis.org

6, 5, 7, 4, 3, 3, 1, 1, 1, 0, 1, 8, 5, 3, 2, 8, 1, 9, 6, 7, 3, 4, 5, 8, 3, 1, 6, 7, 6, 8, 0, 8, 6, 8, 4, 1, 1, 6, 8, 5, 3, 4, 4, 1, 0, 6, 6, 3, 5, 3, 9, 8, 1, 6, 1, 0, 5, 0, 4, 3, 9, 2, 6, 3, 4, 6, 1, 3, 8, 7, 3, 8, 7, 3, 7, 1, 8, 5, 2, 6, 8, 0, 3, 4, 7, 8, 2

Offset: 2

Amiram Eldar, Oct 20 2020

The sum of the reciprocals of the terms of the complement of A171102: numbers with at most 9 distinct digits. It is the union of the 10 sequences of numbers without a single given digit (see the Crossrefs section).

The terms in the data section were taken from the 200 decimal digits given by Strich and Müller (2020).

			65.74331110185328196734583167680868411685344106635398...

Robert Strich and Eric Müller, Ziffernreduzierte Zahlen, in: E. Specht, E. Quaisser and P. Bauermann (eds.), 50 Jahre Bundeswettbewerb Mathematik, Springer Spektrum, Berlin, Heidelberg, 2020, pp. 171-182.

Cf. A171102, A179954, A082830, A082831, A082832, A082833, A082834, A082835, A082836, A082837, A082838, A082839.

Cf. A052382 (numbers without the digit 0), A052383 (without 1), A052404 (without 2), A052405 (without 3), A052406 (without 4), A052413 (without 5), A052414 (without 6), A052419 (without 7), A052421 (without 8), A007095 (without 9).

Equals 1/1 + 1/2 + 1/3 + ... + 1/1023456788 + 1/1023456790 + ..., i.e., A171102(1) = 1023456789 is the first number whose reciprocal is not in the sum.

A050278 Pandigital numbers: numbers containing the digits 0-9. Version 1: each digit appears exactly once.

Original entry on oeis.org

1023456789, 1023456798, 1023456879, 1023456897, 1023456978, 1023456987, 1023457689, 1023457698, 1023457869, 1023457896, 1023457968, 1023457986, 1023458679, 1023458697, 1023458769, 1023458796, 1023458967, 1023458976, 1023459678, 1023459687, 1023459768

Offset: 1

Eric W. Weisstein, Dec 11 1999

This is a finite sequence with 9*9! = 3265920 terms: a(9*9!) = 9876543210.
A171102 is the infinite version, where each digit must appear at least once.
More precisely, this is exactly the subset of the first 9*9! terms of A171102. - M. F. Hasler, Jan 05 2020
Subsequence of A134336 and of A178403; A178401(a(n)) = 1. - Reinhard Zumkeller, May 27 2010
Smallest prime factors: A178775(n) = A020639(a(n)). - Reinhard Zumkeller, Jun 11 2010
A178788(a(n)) = 1. - Reinhard Zumkeller, Jun 30 2010
All these numbers are composite because the sum of the digits, 45, is divisible by 9. - T. D. Noe, Nov 09 2011
This is the 10th row of the array T(k,n) = n-th number in which the number of distinct base-10 digits is k. A031969 is the 4th row. A220063 is the 5th row. A220076 is the 6th row. A218019 is the 7th row. A219743 is the 8th row. - Jonathan Vos Post, Dec 05 2012
From Hieronymus Fischer, Feb 13 2013: (Start)
The sum of all terms is 9!*49444444440 = 17942399998387200.
General formula for the sum of all terms of the finite sequence of the corresponding base-p pandigital numbers with p places: sum = ((p^2 - p - 1)*(p^p - 1) + p - 1)*(p-2)!/2.
General formula for the sum of all terms (interpreted as decimal permutational numbers with exactly d+1 different digits from the range 0..d < 10): sum = (d+1)!*((10d - 1)*10^d - d + 1)/18, d > 1.
(End)

Robert G. Wilson v, Table of n, a(n) for n = 1..1000
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. A171102, A050288, A050289.
Cf. A199630, A199631, A114260, A199632, A199633.
Cf. A031969, A050278, A220063, A220076, A218019, A219743.

Select[ FromDigits@# & /@ Permutations[ Range[0, 9]], # > 10^9 &, 20] (* Robert G. Wilson v, May 30 2010, Jan 17 2012 *)

A050278(n)={ my(b=vector(9,k,1+(n+9!-1)%(k+1)!\k!), t=b[9]-1, d=vector(9,i,i+(i>t)-1)); for(i=1,8, t=10*t+d[b[9-i]]; d=vecextract(d,Str("^"b[9-i]))); t*10+d[1]} \\ M. F. Hasler, Jan 15 2012

is_A050278(n)={ 9<#vecsort(Vecsmall(Str(n)),,8) & n<1e10 } /* assuming that n is a nonnegative integer */ /* M. F. Hasler, Jan 10 2012 */

a(n)=my(d=numtoperm(10,n+9!-1));sum(i=1,#d,(d[i]-1)*10^(#d-i)) \\ David A. Corneth, Jun 01 2014

from itertools import permutations
A050278_list = [int(''.join(d)) for d in permutations('0123456789',10) if d[0] != '0'] # Chai Wah Wu, May 25 2015

A050278 = 9*A171571. - M. F. Hasler, Jan 12 2012

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

Edited by N. J. A. Sloane, Sep 25 2010 to clarify that this is a finite sequence

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

A050288 Pandigital primes.

Original entry on oeis.org

10123457689, 10123465789, 10123465897, 10123485679, 10123485769, 10123496857, 10123547869, 10123548679, 10123568947, 10123578649, 10123586947, 10123598467, 10123654789, 10123684759, 10123685749, 10123694857, 10123746859, 10123784569, 10123846597, 10123849657, 10123854679

Offset: 1

Eric W. Weisstein

Digits may appear multiple times; density n/log n (almost all primes are pandigital).

Note that actually a(n) is much larger than n*log(n) (see Formula section). Even for n = 10000, a(n) = 111571*n*log(n). - Zak Seidov, Jul 27 2014

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000.
Eric Weisstein's World of Mathematics, "Pandigital Number".

Cf. A050278.

ta={{0}};Do[u=Union[IntegerDigits[n]]; If[Equal[u, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}]&&PrimeQ[n], ta=Append[ta, n]], {n, 10123456789, 20000000000}];ta (* Labos Elemer *)

is(n)=isprime(n) && #vecsort(digits(n),,8)>9 \\ Charles R Greathouse IV, May 04 2013

from sympy import isprime
from itertools import count, islice, product
def agen(): # generator of terms
    for d in count(11):
        for f in "123456789":
            for m in product("0123456789", repeat=d-2):
                for e in "1379":
                    t = f + "".join(m) + e
                    if len(set(t)) == 10 and isprime(it:=int(t)):
                        yield it
print(list(islice(agen(), 20))) # Michael S. Branicky, Apr 09 2024

a(n) ~ n log n. - Charles R Greathouse IV, Sep 14 2012

Intersection of A171102 and A000040. - Charles R Greathouse IV, May 04 2013

cp's OEIS Frontend

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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A180489 Smallest pandigital number (A171102) divisible by the n-th prime A000040(n).

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A175845 Pandigital numbers (A171102) of the form 9*prime.

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A276510 Numbers k such that the sum of all the different permutations of the digits of k (A045876(k)) is a pandigital number (a term of A171102).

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A292569 Least pandigital number which sums up with the n-th pandigital number A171102(n) to another pandigital number.

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A292570 Least k > 0 such that A171102(n) + A171102(k) is again a term of A171102, the pandigital numbers (having each digit from '0' to '9' at least once).

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A338287 Decimal expansion of the sum of reciprocals of the numbers that are not pandigital numbers (version 2, A171102).

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A050278 Pandigital numbers: numbers containing the digits 0-9. Version 1: each digit appears exactly 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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