A171102 -id:A171102 - cp's OEIS frontend

A260217 Number of base-3 n-digit pandigital numbers.

0, 0, 4, 24, 100, 360, 1204, 3864, 12100, 37320, 114004, 346104, 1046500, 3155880, 9500404, 28566744, 85831300, 257756040, 773792404, 2322425784, 6969374500, 20912317800, 62745342004, 188252803224, 564791964100, 1694443001160, 5083463221204, 15250658099064

Offset: 1

Jon E. Schoenfield, Jul 19 2015

From Manfred Boergens, Aug 02 2023: (Start)
a(n+1) is the number of pairs (A,B) where A and B are nonempty subsets of {1,2,...,n} and one of these subsets is a proper subset of the other.
If "proper" is omitted, see A091344.
If empty subsets are included, see A027649 (all subsets) and A056182 (proper subsets). (End)

			a(3)=4 because, in base 3, there are four 3-digit pandigital numbers (11=102_3, 15=120_3, 19=201_3, and 21=210_3).
a(4)=24 because, in base 3, there are 24 4-digit pandigital numbers (1002_3, 1012_3, 1020_3, 1021_3, 1022_3, 1102_3, 1120_3, 1200_3, 1201_3, 1202_3, 1210_3, 1220_3, 2001_3, 2010_3, 2011_3, 2012_3, 2021_3, 2100_3, 2101_3, 2102_3, 2110_3, 2120_3, 2201_3, and 2210_3).

Svenja Huntemann, Values, Temperatures, and Enumeration of Placement Games, Slides, Alberta-Montana Combinatorics and Algorithms Day, Banff, Canada, 23-25 June 2023. See p. 105/109.
Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Cf. A000392, A027649, A028243, A056182, A091344, A171102.

[2*3^(n-1) - 2^(n+1) + 2: n in [1..30]]; // Vincenzo Librandi, Jul 20 2015

Table[2 3^(n - 1) - 2^(n + 1) + 2, {n, 30}] (* Vincenzo Librandi, Jul 20 2015 *)

a(n) = 2*A028243(n) = 2*3^(n-1) - 2^(n+1) + 2.
a(n) = 4*A000392(n).
G.f.: 4*x^3/((1-x)*(1-2*x)*(1-3*x)).
E.g.f.: 2/3*((exp(x)-1)^3).

A277054 Least k such that n-th repunit times k is a pandigital.

Original entry on oeis.org

1023456789, 93125079, 9222117, 1110789, 92115, 92115, 92115, 92115, 92115, 92115, 92115, 92115, 92115, 92115, 92115, 92115, 92115, 92115, 92115, 92115, 92115, 92115, 92115, 92115, 92115, 92115, 92115, 92115, 92115, 92115, 92115, 92115, 92115, 92115, 92115

Offset: 1

Andrey Zabolotskiy and Altug Alkan, Sep 26 2016

Starting from n=5, a(n)=A277056(10)=92115, and the corresponding pandigitals are 1023499...989765 (n-5 nines).
a(1)-a(5) constitute row 10 of A277055.
a(n)*A002275(n) is 1023456789, 1024375869, 1023654987, 1234086579, 1023489765, ...

			a(2) = 93125079 because A002275(2)*93125079 = 11*93125079 = 1024375869 that is a pandigital and 93125079 is the least number with this property.

Cf. A002275, A171102, A277055, A277056, A277057.

A277055 Irregular array by rows: A(n,m) is the least number which gives a pandigital product when multiplied by the m-th repunit in base n; each row is truncated when it reaches its stationary point.

Original entry on oeis.org

2, 11, 8, 5, 75, 15, 7, 694, 119, 34, 8345, 1505, 195, 123717, 105803, 2217, 2134, 727, 2177399, 241934, 37303, 3724, 44317196, 4431858, 487068, 54771, 9124, 1023456789, 93125079, 9222117, 1110789, 92115, 26432593615

Offset: 2

Andrey Zabolotskiy and Altug Alkan, Sep 26 2016

For row n, the initial number is A049363(n) and the trailing number is A277056(n). Row 10 is A277054(1)-A277054(5) [note that the initial row is row 2].

			The first rows of the array are:
2, (2, 2...)
11, 8, 5, (5, 5...)
75, 15, 7, (7, 7...)
694, 119, 34,
8345, 1505, 195,
123717, 105803, 2217, 2134, 727,
2177399, 241934, 37303, 3724,
44317196, 4431858, 487068, 54771, 9124,
1023456789, 93125079, 9222117, 1110789, 92115

Cf. A002275, A171102, A049363, A277054, A277056, A277058.

A277056 Least k such that any sufficiently long repunit multiplied by k is a pandigital number in numerical base n.

Original entry on oeis.org

2, 5, 7, 34, 195, 727, 3724, 9124, 92115, 338161, 2780514, 6871290, 99000993

Offset: 2

Andrey Zabolotskiy and Altug Alkan, Sep 26 2016

Trailing terms of rows of A277055.

Written in base n, the terms read: 10, 12, 13, 114, 523, 2056, 7214, 13457, 92115, 21107A, B21116, 156776A, D211117, ...

			Any binary repunit multiplied by 2 is a binary pandigital, so a(2)=2 (10 in binary).
k-th decimal repunit for k>4 multiplied by 92115 gives a decimal pandigital number (see A277054) with no number less than 92115 having the same property, so a(10)=92115.

Cf. A002275, A171102, A277054, A277055, A277059.

Conjecture: for even n>4, a(n) = (n-2)*n^(n/2-1) + n^(n/2-2) + (n^(n/2)-1)/(n-1) + n/2 - 1.

A337127 Table with 10 columns read by rows: T(n, k) is the number of n-digit positive integers with exactly k distinct base 10 digits (0 < k <= 10).

Original entry on oeis.org

9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 81, 0, 0, 0, 0, 0, 0, 0, 0, 9, 243, 648, 0, 0, 0, 0, 0, 0, 0, 9, 567, 3888, 4536, 0, 0, 0, 0, 0, 0, 9, 1215, 16200, 45360, 27216, 0, 0, 0, 0, 0, 9, 2511, 58320, 294840, 408240, 136080, 0, 0, 0, 0, 9, 5103, 195048, 1587600, 3810240, 2857680, 544320, 0, 0, 0

Offset: 1

Stefano Spezia, Aug 17 2020

			The table T(n, k) begins:
9     0      0       0       0       0  0  0  0  0
9    81      0       0       0       0  0  0  0  0
9   243    648       0       0       0  0  0  0  0
9   567   3888    4536       0       0  0  0  0  0
9  1215  16200   45360   27216       0  0  0  0  0
9  2511  58320  294840  408240  136080  0  0  0  0
...

Index entries for sequences related to digits.

Cf. A010785, A031955, A031962, A031969, A031987, A116670, A171102, A218019, A219743, A220076.

Cf. A010734, A048993, A052268 (row sums), A073531 (diagonal), A180599 (k = 1), A335843 (k = 2), A337313 (k = 3).

T[n_,k_]:=9Pochhammer[11-k,k-1]/k!*n!*Coefficient[Series[(Exp[x]-1)^k,{x,0,n}],x,n]; Table[T[n,k],{n,7},{k,10}]//Flatten

T(n, k) = 9*Pochhammer(11-k, k-1)*n! * [x^n] (exp(x) - 1)^k/k!.
T(n, k) = 9*Pochhammer(11-k, k-1) * [x^n] x^k/Product_{j=1..k} (1-j*x).
T(n, k) = 9*Pochhammer(11-k, k-1)*S2(n, k) where S2(n, k) = A048993(n, k) are the Stirling numbers of the 2nd kind.

A358097 a(n) is the smallest integer m > n such that m and n have no common digit, or -1 when such integer m does not exist.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 22, 20, 30, 20, 20, 20, 20, 20, 20, 20, 31, 30, 30, 40, 30, 30, 30, 30, 30, 30, 41, 40, 40, 40, 50, 40, 40, 40, 40, 40, 51, 50, 50, 50, 50, 60, 50, 50, 50, 50, 61, 60, 60, 60, 60, 60, 70, 60, 60, 60, 71, 70, 70, 70, 70, 70, 70, 80, 70, 70, 81, 80, 80, 80, 80

Offset: 0

Bernard Schott, Oct 29 2022

When n is pandigital with or without 0 (A050278, A050289, A171102), m does not exist, so a(n) = -1; see examples for smallest pandigital cases.

			a(10) = 22; a(11) = 20; a(12) = 30.
a(123456789) = -1; a(1234567890) = -1.

Michel Marcus, Table of n, a(n) for n = 0..10000

Cf. A002275, A050278, A050289, A171102.
Cf. A030283 (trajectory starting 0).
Cf. A358098 (similar, with largest integer m < n).

a[n_] := Module[{d = Complement[Range[0, 9], IntegerDigits[n]], m = n + 1}, If[d == {} || d == {0}, -1, While[! AllTrue[IntegerDigits[m], MemberQ[d, #] &], m++]; m]]; Array[a, 100, 0] (* Amiram Eldar, Oct 29 2022 *)

isfull(d) = my(dd=setminus([0..9], d)); (dd==[]) || (dd==[0]);
a(n) = my(d=Set(digits(n))); if (isfull(d), -1, my(k=n+1); while (#setintersect(Set(digits(k)), d), k++); k); \\ Michel Marcus, Oct 29 2022

from itertools import count, product
def a(n):
    s = str(n)
    r = sorted(set("1234567890") - set(s))
    if len(r) == 0 or r == ["0"]: return -1
    for d in count(len(s)):
        for p in product(r, repeat=d):
            m = int("".join(p))
            if m > n: return m
print([a(n) for n in range(75)]) # Michael S. Branicky, Oct 29 2022

a(10^n-k) = 10^n when n >= 2 and 1 <= k <= 8.

a(10^n) = 2 * A002275(n+1), when n >= 1.

A217110 Number of pandigital numbers with n places.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 3265920, 179625600, 5568393600, 128432304000, 2458427811840, 41355201888000, 632788296940800, 9008498667168000, 121205358007493760, 1558813928579107200, 19326359087766057600, 232491479092720848000, 2727512837264447527680, 31331281164921975283200, 353549170783043484480000

Offset: 1

Hieronymus Fischer, Feb 13 2013

The number of numbers between 10^(n-1) and 10^n which contain all decimal digits 0..9.
The ratio a(n)/(10^n-10^(n-1)) indicates the relative proportion of pandigital n-digit numbers compared to all n-digit numbers. Since that ratio converges to the limit 1 for n->oo this can be expressed for large numbers as follows (in a slightly popular manner): "Almost all numbers contain all decimal digits 0..9".
Example: a(n)/(10^n-10^(n-1)) = 0.99973439517775... for n = 100; in this case 99.9734...% of all 100-digit numbers contain all digits 0..9. Conversely, only the tiny proportion of 0.00026560482224... (< 0.03%) lacks one digit. That's astonishing! Intuitively, this is not what one would expect. In fact, for smaller numbers (with which most people are faced normally) the relative portion of numbers which missing at least one digit is significantly larger. Of course, for n < 10 the portion is 100%, and even for numbers with n = 10 or 20 digits the relative proportion of numbers which do not contain all digits 0..9 is 99.96371...% or 78.52626...%, respectively. The least number of digits for which the pandigital numbers hold the majority is 27. Here, the proportion of numbers which do not contain all digits is 48.03664...%. So one could bet that a randomly chosen number with >= 27 digits contains all digits.

			a(k) = 0 for k < 10 since there are no pandigital numbers with < 10 places, trivially.
a(10) = 9*9! since the first digit can be in the range 1..9 and for the following 9 digits there are 9, 8, 7, ..., 1 possible values.

Hieronymus Fischer, Table of n, a(n) for n = 1..200

Cf. A171102, A050278, A011540, A002542, A053283, A217094.

a(n) = 9*9!*S2(n,10), where the S2(n,10) are the Stirling numbers of the second kind (cf. triangle A008277).
Asymptotic behavior: Limit_{n->oo} a(n)/10^n = 9/10.
G.f.: g(x) = 9*9!*x^10/(Product_{j=1..10} (1-jx)).
E.g.f. g(x) = (9/10) * (e^x - 1)^10.

A226273 Number of distinct values u^v, where u and v are digits occurring in decimal representation of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 4, 3, 4, 4, 4, 4, 4, 3, 3, 4, 1, 4, 4, 4, 4, 4, 4, 3, 3, 3, 4, 1, 4, 4, 4, 4, 4, 3, 3, 4, 4, 4, 1, 4, 4, 4, 4, 3, 3, 4, 4, 4, 4, 1, 4, 4, 4, 3, 3, 4, 4, 4, 4, 4, 1, 4, 4, 3, 3, 4, 4, 4, 4

Offset: 0

Reinhard Zumkeller, Jul 09 2013

Row lengths in table A226272;
a(n) <= 61; A171102(1) = 1023456789 is the smallest number m such that a(m) = 61.
a(n) = 61 for almost all n, in the sense that the natural density of n such that a(n) < 61 is 0. - Charles R Greathouse IV, May 13 2015

			See A226272.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

a226273 = length . a226272_row :: Integer -> Int

a(n)=if(n<9, return(1)); my(d=Set(digits(n)),v=List()); for(i=1,#d, for(j=1,#d, listput(v,d[i]^d[j]))); #Set(v) \\ Charles R Greathouse IV, May 13 2015

A262277 Numbers having in decimal representation the same distinct decimal digits as their 9's complement.

Original entry on oeis.org

18, 27, 36, 45, 54, 63, 72, 81, 90, 118, 181, 188, 227, 272, 277, 336, 363, 366, 445, 454, 455, 544, 545, 554, 633, 636, 663, 722, 727, 772, 811, 818, 881, 900, 909, 990, 1089, 1098, 1118, 1181, 1188, 1278, 1287, 1368, 1386, 1458, 1485, 1548, 1584, 1638

Offset: 1

Reinhard Zumkeller, Sep 17 2015

If d is a digit of any term then also 9 - d;

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A061601, A227362, subsequences: A111708, A050278, A171102.

import Data.List (nub, sort)
a262277 n = a262277_list !! (n-1)
a262277_list = filter f [1..] where
   f x = sort ds' == sort (map (9 -) ds') where
     ds' = nub $ ds x
     ds 0 = []; ds z = d : ds z' where (z', d) = divMod z 10

isok(m) = my(d=digits(m), c=apply(x->9-x, d)); Set(d) == Set(c); \\ Michel Marcus, Jan 22 2022

A227362(A061601(a(n))) = A227362(a(n)).

A272269 Numbers n such that 11^n does not contain all ten decimal digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 25, 27, 28, 34, 38, 41

Offset: 1

Altug Alkan, Apr 24 2016

Inspiration was the simple form of 11 that is concatenation of 1 and 1. With similar motivation, A130696 focuses on the values of 2^n = (1 + 1)^n. Since this sequence exists in base 10, 11^n*10 is simply concatenation of 11^n and 0. So 11^(n+1) = concat(11^n, 0) + 11^n while 2^(n+1) = 2^n + 2^n.

A030706 is a subsequence. So note that if there is currently no proof of finiteness of A030706, then there is no proof yet of the finiteness of this sequence.

			25 is a term because 11^25 = 108347059433883722041830251 that does not contain digit 6.
26 is not a term because 11^26 = 11^25*10 + 11^25 = 1083470594338837220418302510 + 108347059433883722041830251 = 1191817653772720942460132761 that contains all ten decimal digits.

Cf. A001020, A030706, A130696, A171102.

Select[Range[0, 120], AnyTrue[DigitCount[11^#], # == 0 &] &] (* Michael De Vlieger, Apr 24 2016, Version 10 *)

isA171102(n) = 9<#vecsort(Vecsmall(Str(n)), , 8);
lista(nn) = for(n=0, nn, if(!isA171102(11^n), print1(n, ", ")));

select( is_A272269(n)=#Set(digits(11^n))<10 ,[0..100]) \\ M. F. Hasler, May 18 2017

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A260217 Number of base-3 n-digit pandigital numbers.

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A277054 Least k such that n-th repunit times k is a pandigital.

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A277055 Irregular array by rows: A(n,m) is the least number which gives a pandigital product when multiplied by the m-th repunit in base n; each row is truncated when it reaches its stationary point.

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A277056 Least k such that any sufficiently long repunit multiplied by k is a pandigital number in numerical base n.

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A337127 Table with 10 columns read by rows: T(n, k) is the number of n-digit positive integers with exactly k distinct base 10 digits (0 < k <= 10).

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A358097 a(n) is the smallest integer m > n such that m and n have no common digit, or -1 when such integer m does not exist.

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A217110 Number of pandigital numbers with n places.

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A226273 Number of distinct values u^v, where u and v are digits occurring in decimal representation of n.

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A262277 Numbers having in decimal representation the same distinct decimal digits as their 9's complement.

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