A171102 -id:A171102 - cp's OEIS frontend

A308377 "Autotomy numbers" that have exactly 10 distinct decimal digits. Subtracting their last digit from the remaining part produces a shorter autotomy number (still with no duplicate digit). This process is iterated until the remaining part has only one digit (details in the Example section).

2487159630, 2581740963, 3697512840, 3751908642, 3791508642, 3796512840, 4283716590, 4573921680, 4609785321, 4832716590, 4960785321, 4976853210, 5016793284, 5071693284, 5106793284, 5170693284, 5179386420, 5187429630, 5389710642, 5397186420, 5473921680, 5710693284, 5731908642, 5786413290, 5791308642, 5809764321, 5839710642, 5847102963, 5897130642, 5897643210, 5907864321

Offset: 1

Eric Angelini and Jean-Marc Falcoz, May 23 2019

The sequence is finite and has 182 terms; a(182) = 9876543210.

			a(2) = 2581740963
Subtract 3 (last digit) from the remaining part 258174096 = 258174093
Subtract 3 (last digit) from the remaining part 25817409 = 25817406
Subtract 6 (last digit) from the remaining part 2581740 = 2581734
Subtract 4 (last digit) from the remaining part 258173 = 258169
Subtract 9 (last digit) from the remaining part 25816 = 25807
Subtract 7 (last digit) from the remaining part 2580 = 2573
Subtract 3 (last digit) from the remaining part 257 = 254
Subtract 4 (last digit) from the remaining part 25 = 21
Subtract 1 (last digit) from the remaining part 2 = 1 (single digit, end).

Jean-Marc Falcoz, Table of n, a(n) for n = 1..182
Eric Angelini, Pandigitaux et saucissons (in French).

Cf. A308393 (definition of an "autotomy number"), A050278 (pandigital numbers, version 1: each digit appears exactly once), A171102 (pandigital numbers, version 2: each digit appears at least once).

A330562 Positive numbers k with property that if d is any nonzero digit of k then k mod d is also a digit of k.

Original entry on oeis.org

10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 101, 102, 103, 104, 105, 109, 110, 120, 130, 140, 150, 190, 200, 201, 202, 204, 206, 208, 210, 220, 230, 240, 250, 260, 280, 290, 300, 301, 302, 303, 306, 309, 310, 320, 330, 360, 390, 400, 401, 402, 404, 408, 420, 440, 460, 480, 500, 501, 502, 504, 505, 510, 520, 540, 550, 590

Offset: 1

N. J. A. Sloane, Dec 31 2019, following a suggestion from Eric Angelini

Theorem: k must have a zero digit.
Proof: If not, let s be the smallest digit in k. Then d = (k mod s) is a digit of k, and d < s. Contradiction.
Pandigital numbers (A171102) are necessarily an infinite subset. - Hans Havermann, Jan 02 2020

			401 is a term since 401 mod 4 = 1 and 401 mod 1 = 0, and 1 and 0 are both digits of 401.

Rémy Sigrist, Table of n, a(n) for n = 1..25000

Cf. A330563 (primes), A171102 (pandigital subset).

[k:k in [1..600]| forall{c:c in Set(Intseq(k)) diff {0}| k mod c in Intseq(k)}]; // Marius A. Burtea, Jan 01 2020

Select[Range@ 600, Function[{k, d}, AllTrue[DeleteCases[d, 0], ! FreeQ[d, Mod[k, #]] &]] @@ {#, IntegerDigits[#]} &] (* Michael De Vlieger, Jan 01 2020 *)

is(k) = my (d=Set(digits(k))); for (i=1, #d, if (d[i] && setsearch(d, k%d[i])==0, return (0))); return (1) \\ Rémy Sigrist, Jan 01 2020

def ok(n): s = set(map(int, str(n))); return all(n%d in s for d in s-{0})
print([k for k in range(1, 600) if ok(k)]) # Michael S. Branicky, Dec 23 2024

A330981 Remodd numbers: having an odd remainder modulo all of their digits, digit 0 forbidden.

Original entry on oeis.org

43, 47, 73, 87, 223, 227, 253, 267, 283, 289, 337, 343, 349, 367, 379, 397, 433, 439, 463, 467, 469, 477, 489, 493, 523, 553, 583, 643, 647, 649, 669, 673, 677, 687, 689, 733, 747, 787, 799, 823, 827, 829, 849, 853, 869, 883, 887, 889, 943, 997

Offset: 1

Eric Angelini and M. F. Hasler, Jan 05 2020

No term may have a digit 0 or 1, therefore the asymptotic density is zero and would be so even if the definition is changed to "digits 0 are allowed but ignored", since pandigital numbers A171102 have asymptotic density 1.
Does not contain any remeven number (A330982), thus in particular none of A010785 (repdigits) or its superset A034838 (divisible by all digits) or A014263 (only even digits). Also no multiples of 2 or 5 (A005843 or A008587) which are even modulo the last digit (unless it is 0), so all terms end in 3, 7 or 9.
Contains the infinite subsequence (43, 433, 4333, ...), but after {47, 477, 4777} not 47777 = 6825*7 + 2, and after {73, 733} not 7333 = 1047*7 + 4, and after {87, 887} not 8887 = 1269*7 + 4.
The first term which contains the digits 2..9 is a(784795) = 224567983. - Giovanni Resta, Jan 07 2020

			43 is in the sequence because 43 mod 4 = 3 and 43 mod 3 = 1 both are odd.

Giovanni Resta, Table of n, a(n) for n = 1..10000
Eric Angelini, Remeven numbers, SeqFan list, Jan 05 2020.

Cf. A330982 (remeven numbers).

Cf. A171102 (pandigitals), A010785 (repdigits), A014263 (only even digits), A034838 (divisible by all digits).

[k:k in [1..1000]|not 0 in Intseq(k) and forall{d:d in Intseq(k)|IsOdd(k mod d)}]; // Marius A. Burtea, Jan 07 2020

select( {is(n, d=Set(digits(n)))=d[1]&&!for(j=1,#d, bittest(n%d[j],0)||return)}, [1..2000])

A330982 Remeven numbers: having an even remainder modulo any of their digits, digit 0 forbidden.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 14, 15, 16, 18, 22, 24, 26, 28, 32, 33, 35, 36, 38, 42, 44, 46, 48, 52, 54, 55, 62, 64, 66, 68, 72, 74, 76, 77, 82, 84, 86, 88, 92, 94, 96, 98, 99, 111, 112, 113, 114, 115, 116, 118, 119, 122, 124, 126, 128, 131, 132, 134, 135, 137, 138, 142, 144, 146, 148, 152

Offset: 1

Eric Angelini and M. F. Hasler, Jan 05 2020

The sequence is a subset of the zeroless numbers A052382 which have asymptotic density 0 because they are in the complement of pandigital numbers A171102 which have asymptotic density 1. But does it have finite density within A052382?
It contains all repdigit numbers A010785 \ {0} and also all numbers with only even digits A014263 \ {0} and all numbers divisible by all of their digits, A034838.
The graph is self-similar, it looks the same whether we take the graph of values < 10^4 or that of values < 10^5 etc.: In the range 0 < a(n) < 10^(k+1), there are jumps of size > 10^k/9 where the values cross the limits d*10^k, 1 <= d <= 9 (from a(n) <= {d-1}9...9 to a(n+1) >= d1...1, since 0's are forbidden).
There are N = (0, 9, 48, 303, 2190, 15871, 119442, 930324, ...) terms below 10^k, k >= 0; these N(k) are also the indices of terms a(N(k)) = 10^k-1 (k>0), which are followed by repunits a(N(k)+1) = a(N(k+1))/9 (k >= 0).
The smallest zeroless pandigital term is a(8455060) = 123567894. - Giovanni Resta, Jan 08 2020

			12 is in the sequence because 12 % 1 = 0 and 12 % 2 = 0 both are even, where x % y is the remainder of x divided by y.
13 is not in the sequence because 13 % 3 = 1 is odd.

M. F. Hasler, Table of n, a(n) for n = 1..10000, Jan 07 2020.
Eric Angelini, Remeven numbers, SeqFan list, Jan 05 2020.

Cf. A330981 (remodd numbers).

Cf. A171102 (pandigitals), A010785 (repdigits), A014263 (only even digits), A034838 (divisible by all digits).

[k:k in [1..160]|not 0 in Intseq(k) and forall{d:d in Intseq(k)|IsEven(k mod d)}]; // Marius A. Burtea, Jan 08 2020

Select[Range[200],DigitCount[#,10,0]==0&&AllTrue[Mod[#,IntegerDigits[ #]],EvenQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 02 2020 *)

select( {is_A330982(n,d=digits(n))=vecmin(d)&&!for(j=1,#d, bittest(n%d[j],0)&&return)}, [1..200]) \\ Using Set(digits) is about 20% slower

A358098 a(n) is the largest 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

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 8, 19, 9, 19, 19, 19, 19, 19, 19, 19, 18, 29, 29, 19, 29, 29, 29, 29, 29, 29, 28, 39, 39, 39, 29, 39, 39, 39, 39, 39, 38, 49, 49, 49, 49, 39, 49, 49, 49, 49, 48, 59, 59, 59, 59, 59, 49, 59, 59, 59, 58, 69, 69, 69, 69, 69, 69, 59, 69, 69, 68, 79

Offset: 1

Bernard Schott, Oct 29 2022

Note that only when n is pandigital with 0 (A050278, A171102), such m does not exist and a(n) = -1; see examples for smallest pandigital cases.

			a(19) = 8, a(20) = 19; a(21) = 9.
a(123456789) = 0; a(1234567890) = -1.

Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A050289, A050278, A171102.

Cf. A358097 (similar, with smallest integer m > n).

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

a(n) = my(d=Set(digits(n))); forstep (m=n-1, 0, -1, if (!#setintersect(d, Set(digits(m))), return(m))); return(-1); \\ Michel Marcus, Oct 30 2022

from itertools import product
def a(n):
    s = str(n)
    r = sorted(set("1234567890") - set(s), reverse=True)
    if len(r) == 0: return -1
    if r == ["0"]: return 0
    for d in range(len(s), 0, -1):
        for p in product(r, repeat=d):
            m = int("".join(p))
            if m < n: return m
print([a(n) for n in range(1, 81)]) # Michael S. Branicky, Oct 29 2022

a(10^n) = 10^n - 1 for n >= 0.

a(A050289(n))=0.

A363927 Numbers N such that in the concatenation of N^2 and N^3, each of the 10 decimal digits appears equally often.

Original entry on oeis.org

69, 6534, 497375, 539019, 543447, 586476, 589629, 601575, 646479, 858609, 895688, 959097, 46839081, 47469378, 47693199, 47760623, 47841576, 48038964, 48527792, 48733506, 48886836, 48965892, 49229103, 49397283, 49594832, 49670616, 50013116, 50247423, 50359157

Offset: 1

Charles R Greathouse IV and M. F. Hasler, Jun 28 2023

a(3) = 497375 and a(11) = 895688 are the only terms < 10^6 that are not divisible by 3.
Each term has an even number of decimal digits, k, and a corresponding value between 10^(k-1)*100^(1/3) and 10^k. - Michael S. Branicky, Jun 29 2023
Indeed, the number of digits of concat(N^2, N^3) is floor(2*L + 1) + floor(3*L + 1) where L = log_10(N). This is a multiple of 10 iff L mod 2 is in the interval [5/3, 2), which means that N is in the above range for some even k. - M. F. Hasler, Jul 02 2023

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A363905, A363909: concat(n^2, n^3) has each digit at least once / twice.
Cf. A171102: pandigital numbers.
Cf. A036744, A054038, A071519 and A156977 for "pandigital squares".
Cf. A119735: n^3 is pandigital.

fQ[n_] := Length@ Union[ Count[ Sort[ Join[ IntegerDigits[n^2], IntegerDigits[n^3]]], #] & /@ Range[0, 9]] == 1; Select[ Range@ 52000000, fQ] (* Robert G. Wilson v, Jul 01 2023 *)

is(n)={my(v=concat(digits(n^2),digits(n^3)), c=#v); c%10==0 && vecsort(v)==[0..c-1]\(c\10)}
for(n=1,1e6, is(n)&& print1(n","))

a(13) and beyond from Michael S. Branicky, Jun 28 2023

A113639 Pandigital strong pseudoprimes (base-2).

Original entry on oeis.org

13069482857, 16974853201, 18260734519, 23759084161, 24785139601, 25089467413, 29384567041, 38706945421, 41390257861, 49065821737, 65829130417, 67903858241, 76953210841, 98642074513, 100695314287, 106237698451, 106751489293, 110586343297, 113646829507, 116548907329

Offset: 1

Shyam Sunder Gupta, Jan 15 2006

			a(1) = 13069482857 is a term because 13069482857 is a strong pseudoprime (base-2) and also contains all digits from 0 to 9 so pandigital also.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A171102 and A001262.

More terms from Amiram Eldar, Nov 10 2019

A113642 Pandigital pseudoprimes (base-2).

Original entry on oeis.org

12539670481, 13069482857, 15076432489, 15342679801, 16974853201, 17392546081, 18260734519, 18394765201, 21654091837, 23759084161, 24785139601, 25089467413, 26714530189, 27631845901, 29384567041, 29673748105, 32798165401, 35901682741, 37946180251, 38204791561

Offset: 1

Shyam Sunder Gupta, Jan 15 2006

			a(1) = 12539670481 is a term because 12539670481 is a pseudoprime (base-2) and also contains all of the digits from 0 to 9 so it's pandigital.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A171102 and A001567.

More terms from Amiram Eldar, Nov 10 2019

Example edited by Harvey P. Dale, Jun 03 2023

A113643 Pandigital Carmichael numbers.

Original entry on oeis.org

17392546081, 18394765201, 32798165401, 61280451937, 96112743805, 103964580721, 129386974501, 165274930081, 176962430851, 189147932065, 194687539201, 206238795841, 236791540801, 248979435601, 249817563601, 294153967801, 334797586201, 343906758721, 384073572961

Offset: 1

Shyam Sunder Gupta, Jan 15 2006

			a(1)=17392546081 is a term because 17392546081 is a Carmichael number and also contain all digits from 0 to 9 so pandigital also.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A002997 and A171102.

More terms from Amiram Eldar, Sep 19 2019

A217111 Number of pandigital numbers <= 10^n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 3265920, 182891520, 5751285120, 134183589120, 2592611400960, 43947813288960, 676736110229760, 9685234777397760, 130890592784891520, 1689704521363998720, 21016063609130056320, 253507542701850904320, 2981020379966298432000

Offset: 1

Hieronymus Fischer, Feb 13 2013

The number of numbers with <= n digits which contain all decimal digits 0..9.
The ratio a(n)/10^n indicates the relative proportion of pandigital numbers <= 10^n compared to all numbers <= 10^n. 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 = 0. 99973107526479... for n = 100; in this case 99.9731...% of all numbers <= 10^100 contain all digits 0..9. Conversely, only the tiny proportion of 0.000268924735210... (< 0.03%) lacks at least 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 <= 10^10 or <= 10^20 the relative proportion of numbers which do not contain all digits 0..9 is 99.96734...% or 78.98393...%, respectively. 10^27 is the least power of 10 such that the pandigital numbers hold the majority. Here, the proportion of pandigital numbers among all numbers <= 10^27 is 51.50961...%. So one could bet that a randomly chosen number <= 10^27 contains all digits.
Partial sums of A217110.

			a(k) = 0, for k < 10 since there are no pandigital numbers <= 10^9, 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, A217110.

3265920 Accumulate[StirlingS2[Range[25],10]] (* Harvey P. Dale, Oct 16 2022 *)

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

cp's OEIS Frontend

A308377 "Autotomy numbers" that have exactly 10 distinct decimal digits. Subtracting their last digit from the remaining part produces a shorter autotomy number (still with no duplicate digit). This process is iterated until the remaining part has only one digit (details in the Example section).

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A330562 Positive numbers k with property that if d is any nonzero digit of k then k mod d is also a digit of k.

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A330981 Remodd numbers: having an odd remainder modulo all of their digits, digit 0 forbidden.

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A330982 Remeven numbers: having an even remainder modulo any of their digits, digit 0 forbidden.

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A358098 a(n) is the largest 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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A363927 Numbers N such that in the concatenation of N^2 and N^3, each of the 10 decimal digits appears equally often.

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Mathematica

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A113639 Pandigital strong pseudoprimes (base-2).

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A113642 Pandigital pseudoprimes (base-2).

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