A010784 -id:A010784 - cp's OEIS frontend

A178787 Number of numbers <= n having distinct digits in their decimal representation, cf. A010784.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 61, 62, 63, 64, 65, 66

Offset: 0

Reinhard Zumkeller, Jun 30 2010

a(m)<=a(n) for m=9876543210;
a(A010784(n)) = n; a(A109303(n)) = a(A109303(n)-1);
partial sums of A178788.

			a(12) = #{0,1,2,3,4,5,6,7,8,9,10,12} = 12;
a(24) = a(12) + #{13,14,15,16,17,18,19,20,21,23,24} = 23.

R. Zumkeller, Table of n, a(n) for n = 0..25000
Eric Weisstein's World of Mathematics, Digit

a178787 n = a178787_list !! n
a178787_list = scanl1 (+) a178788_list
-- Reinhard Zumkeller, Mar 15 2014

f[n_] := Length[ Select[ Range[0, n], Max[ DigitCount[#]] == 1 &]]; Array[f, 70, 0] (* Robert G. Wilson v, Dec 13 2016 after Harvey P. Dale in A010784 *)

A321149 a(1) = 102735, a(n) = prime(n-1)*a(n-1) but products that are not in A010784 are first reduced as in A320486 (see comments); continue until zero is reached.

Original entry on oeis.org

102735, 2547, 7641, 38205, 267435, 2941785, 8405, 1425, 205, 4715, 1675, 192, 7104, 9164, 394052, 18520, 981560, 579124, 24, 1608, 468, 316, 296, 24568, 186, 18042, 184, 18952, 7864, 8516, 962308, 36

Offset: 1

Hans Havermann, Oct 28 2018

At each step, integers that contain duplicated digits are reduced to terms of A010784 by erasing all digits that appear more than once and bunching up the digits that remain. Leading zeros are ignored and any number that disappears entirely becomes 0. See A320486.

102735 is the smallest of 785 A010784 terms that result in a 362-term sequence, the longest possible.

			2 * 102735 = [205470] => 2547
3 * 2547 = 7641
5 * 7641 = 38205
7 * 38205 = 267435
11 * 267435 = 2941785
13 * 2941785 = [38243205] => 8405
17 * 8405 = [142885] => 1425
19 * 1425 = [27075] => 205
...
2417 * 40 = [96680] => 980
2423 * 980 = [2374540] => 23750
2437 * 23750 = [57878750] => 0

Hans Havermann, Table of n, a(n) for n = 1..362
Hans Havermann, What's so special about 102735?

Cf. A010784, A321148, A320486.

A321148 a(1) = 24603, a(n) = n*a(n-1) but products that are not in A010784 are first reduced as in A320486 (see comments); continue until zero is reached.

Original entry on oeis.org

24603, 49206, 4768, 19072, 95360, 572160, 4512, 309, 2781, 27810, 3591, 43092, 5019, 702, 153, 28, 476, 56, 1064, 180, 3780, 83160, 92680, 430, 175, 40, 18, 504, 4, 120, 3720, 94, 3102, 105468, 69180, 298, 26, 9, 351, 1, 41, 17, 731, 32164, 17380, 7480, 3160, 5680, 7830, 3915, 15, 780, 130, 72, 3960, 1760, 132, 75, 25, 15, 915, 56730, 570, 36480, 371, 286, 962, 541, 729, 513, 642, 6, 438, 341, 27, 5, 385, 0

Offset: 1

Hans Havermann, Oct 28 2018

At each step, integers that contain duplicated digits are reduced to terms of A010784 by erasing all digits that appear more than once and bunching up the digits that remain. Leading zeros are ignored and any number that disappears entirely becomes 0. See A320486.

24603 is the smallest of 1746 A010784 terms that result in a 78-term sequence, the longest possible.

			2 * 24603 = 49206
3 * 49206 = [147618] => 4768
4 * 4768 = 19072
5 * 19072 = 95360
6 * 95360 = 572160
7 * 572160 = [4005120] => 4512
8 * 4512 = [36096] => 309
...
76 * 27 = [2052] => 5
77 * 5 = 385
78 * 385 = [30030] => 0

Hans Havermann, What's so special about 102735?

Cf. A010784, A321149, A320486.

A160343 Numbers k such that the two closest numbers above and below k, which are in A010784 and which have no common digit with k, have the same distance to k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 89, 394, 605, 894, 3944, 6055, 8944, 15111, 84888, 89444, 894444

Offset: 1

Rodolfo Kurchan, May 10 2009, May 11 2009, May 16 2009

For each integer k, define the smallest upper neighbor k+d with d > 0 such that k+d contains each digit at most once (see A010784) and has none of the digits of k. Define also the largest lower neighbor k-b with b > 0 such that k-b contains each digit at most once and has none of the digits of k.
The sequence consists of those k where d=b, that is, where these two neighbors are equidistant from k.
From Donovan Johnson, Sep 29 2009: (Start)
15111 has neighbors 9876 and 20346, distance 5235.
84888 has neighbors 79653 and 90123, distance 5235.
89444 has neighbors 76532 and 102356, distance 12912.
894444 has neighbors 765321 and 1023567, distance 129123.
Sequence is complete.
(End)

			6 has neighbors 5 and 7, common distance 1.
89 has neighbors 76 and 102, common distance 13.
394 has neighbors 287 and 501, distance 107.
605 has neighbors 498 and 712, distance 107.
894 has neighbors 765 and 1023, distance 129.
3944 has neighbors 2876 and 5012, distance 1068.
6055 has neighbors 4987 and 7123, distance 1068.
8944 has neighbors 7653 and 10235, distance 1291.
94 is not in the sequence because 87 and 102 have distances 7 and 8.

Rodolfo Kurchan and Claudio Meller, Snark e-mail list, May 10, 2009 [Dead link]

Edited by R. J. Mathar, May 20 2009

a(17)-a(20) from Donovan Johnson, Sep 29 2009

A043096 Numbers in which every pair of adjacent digits are distinct.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 101, 102, 103, 104, 105, 106, 107, 108, 109, 120, 121, 123, 124, 125, 126, 127, 128, 129

Offset: 1

Clark Kimberling

Superset of A010784. - R. J. Mathar, Oct 20 2008

Complement of A171901; A196368(a(n)) = 1. - Reinhard Zumkeller, Oct 28 2001

			11 is the first number not in the sequence, since it has a pair of identical adjacent digits.

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

Cf. A010784, A171901, A196368.

import Data.List (elemIndices)
a043096 n = a043096_list !! n
a043096_list = elemIndices 1 a196368_list
-- Reinhard Zumkeller, Oct 28 2001

isA034096 := proc(n)
    local dgs ;
    dgs := convert(n,base,10) ;
    for i from 2 to nops(dgs) do
        if op(i,dgs) = op(i-1,dgs) then
            return false;
        end if;
    end do:
    true ;
end proc:
for n from 0 to 150 do
    if isA034096(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Jun 26 2023

t={}; Do[If[!MemberQ[Differences[IntegerDigits[n]], 0], AppendTo[t,n]], {n,0,69}]; t (* Jayanta Basu, May 04 2013 *)

is(n)=my(v=digits(n));for(i=2,#v,if(v[i]==v[i-1],return(0)));1 \\ Charles R Greathouse IV, Sep 17 2012

def ok(n): s = str(n); return all(s[i] != s[i+1] for i in range(len(s)-1))
print(list(filter(ok, range(70)))) # Michael S. Branicky, Aug 22 2021

Offset corrected by Reinhard Zumkeller, Oct 28 2011

A029743 Primes with distinct digits.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 103, 107, 109, 127, 137, 139, 149, 157, 163, 167, 173, 179, 193, 197, 239, 241, 251, 257, 263, 269, 271, 281, 283, 293, 307, 317, 347, 349, 359, 367, 379, 389

Offset: 1

Patrick De Geest

This sequence has 283086 terms, the last being 987654103 = A007810(9). - Jud McCranie

Intersection of A010784 and A000040; A178788(a(n)) * A010051(a(n)) = 1. [Reinhard Zumkeller, Sep 25 2011]

Reinhard Zumkeller, Table of n, a(n) for n = 1..283086, full sequence
Patrick De Geest, World!Of Numbers

Cf. A000040, A007810, A010051, A010784, A178788.

a029743 n = a029743_list !! (n-1)
a029743_list = filter ((== 1) . a010051) a010784_list
-- Reinhard Zumkeller, Sep 25 2011

t={};Do[p=Prime[n];If[Select[Transpose[Tally[IntegerDigits[p]]][[2]],#>1 &]=={},AppendTo[t,p]],{n,77}];t (* Jayanta Basu, May 04 2013 *)
Select[Prime[Range[80]],Max[DigitCount[#]]<2&] (* Harvey P. Dale, Sep 13 2020 *)

from sympy import isprime
from itertools import permutations as P
dist = [p for d in range(1, 11) for p in P("0123456789", d) if p[0] != "0"]
afull = [t for t in (int("".join(p)) for p in dist) if isprime(t)]
print(afull[:100]) # Michael S. Branicky, Aug 04 2022

A178788 Characteristic function of numbers having distinct digits in their decimal representation.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1

Offset: 0

Reinhard Zumkeller, Jun 30 2010

a(A010784(n)) = 1; a(A109303(n)) = 0;
first differences of A178787.
a(n) <= A196368(n).
a(n) = 0 for n > 9*9!. - Hieronymus Fischer, Feb 02 2013

R. Zumkeller, Table of n, a(n) for n = 0..10000
Michael De Vlieger, Plot a(n) at (x,y) = (n mod 1000, -floor(n/1000)), n = 1..10^6, where black represents a(n) = 1 and white represents a(n) = 0.
Eric Weisstein's World of Mathematics, Digit
Index entries for characteristic functions

import Data.List (nub)
a178788 n = fromEnum $ nub (show n) == show n
-- Reinhard Zumkeller, Sep 25 2011

lst = {}; Do[If[Select[Tally[IntegerDigits[n]][[All, 2]], # > 1 &] == {}, AppendTo[lst, 1], AppendTo[lst, 0]], {n, 0, 104}]; lst (* Arkadiusz Wesolowski, May 10 2012 *)

a(n) = 0^(A055642(n)-A043537(n)).

A247802 Numbers in decimal representation with distinct digits, such that in Dutch their digits are in alphabetic order.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 19, 24, 25, 26, 27, 30, 32, 34, 35, 36, 37, 39, 45, 46, 47, 56, 57, 67, 80, 81, 82, 83, 84, 85, 86, 87, 89, 90, 92, 94, 95, 96, 97, 102, 104, 105, 106, 107, 124, 125, 126, 127, 130, 132, 134, 135, 136

Offset: 1

Reinhard Zumkeller, Oct 05 2014

List of decimal digits, alphabetically sorted by their names in Dutch:
8  acht, 1  een, 3  drie, 9  negen, 0  nul, 2  twee, 4  vier, 5  vijf, 6  zes, 7  zeven;
finite sequence with last and largest term a(992) = 8139024567.

Reinhard Zumkeller, Table of n, a(n) for n = 1..992
Wikipedia, Zahlen in unterschiedlichen Sprachen
Wikipedia, List of numbers in various languages

Intersection of A010784 and A247752.

Cf. A247800 (Czech), A247801 (Danish), A053433 (English), A247803 (Finnish), A247804 (French), A247805 (German), A247806 (Hungarian), A247807 (Italian), A247808 (Latin), A247809 (Norwegian), A247810 (Polish), A247807 (Portuguese), A247811 (Russian), A247812 (Slovak), A247813 (Spanish), A247809 (Swedish), A247814 (Turkish).

import Data.IntSet (fromList, deleteFindMin, union)
import qualified Data.IntSet as Set (null)
a247802 n = a247802_list !! (n-1)
a247802_list = 0 : f (fromList [1..9]) where
   f s | Set.null s = []
       | otherwise  = x : f (s' `union`
         fromList (map (+ 10 * x) $ tail $ dropWhile (/= mod x 10) digs))
       where (x, s') = deleteFindMin s
   digs = [8, 1, 3, 9, 0, 2, 4, 5, 6, 7]

A247800 Numbers in decimal representation with distinct digits, such that in Czech their digits are in alphabetic order.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 15, 16, 17, 18, 20, 21, 23, 25, 26, 27, 28, 40, 41, 42, 43, 45, 46, 47, 48, 49, 53, 56, 57, 63, 73, 76, 83, 85, 86, 87, 90, 91, 92, 93, 95, 96, 97, 98, 103, 105, 106, 107, 108, 153, 156, 157, 163, 173, 176, 183, 185, 186

Offset: 1

Reinhard Zumkeller, Oct 05 2014

List of decimal digits, alphabetically sorted by their names in Czech:
4  čtyři, 9  devět, 2  dva/dvě, 1  jeden/jedna/jedno, 0  nula, 8  osm, 5  pět, 7  sedm, 6  šest, 3  tři;
finite sequence with last and largest term a(992) = 4921085673.

Reinhard Zumkeller, Table of n, a(n) for n = 1..992
Wikipedia, Zahlen in unterschiedlichen Sprachen
Wikipedia, List of numbers in various languages

Intersection of A010784 and A247750.

Cf. A247801 (Danish), A247802 (Dutch), A053433 (English), A247803 (Finnish), A247804 (French), A247805 (German), A247806 (Hungarian), A247807 (Italian), A247808 (Latin), A247809 (Norwegian), A247810 (Polish), A247807 (Portuguese), A247811 (Russian), A247812 (Slovak), A247813 (Spanish), A247809 (Swedish), A247814 (Turkish).

import Data.IntSet (fromList, deleteFindMin, union)
import qualified Data.IntSet as Set (null)
a247800 n = a247800_list !! (n-1)
a247800_list = 0 : f (fromList [1..9]) where
   f s | Set.null s = []
       | otherwise  = x : f (s' `union`
         fromList (map (+ 10 * x) $ tail $ dropWhile (/= mod x 10) digs))
       where (x, s') = deleteFindMin s
   digs = [4, 9, 2, 1, 0, 8, 5, 7, 6, 3]

A247801 Numbers in decimal representation with distinct digits, such that in Danish their digits are in alphabetic order.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 23, 40, 42, 43, 46, 47, 48, 49, 50, 52, 53, 54, 56, 57, 58, 59, 62, 63, 67, 72, 73, 82, 83, 86, 87, 90, 92, 93, 96, 97, 98, 102, 103, 106, 107, 108, 123, 140, 142, 143, 146, 147, 148, 149, 150

Offset: 1

Reinhard Zumkeller, Oct 05 2014

List of decimal digits, alphabetically sorted by their names in Danish:
1  en/et, 5  fem, 4  fire, 9  ni, 0  nul, 8  otte, 6  seks, 7  syv, 2  to, 3  tre;
finite sequence with last and largest term a(992) = 1549086723.

Reinhard Zumkeller, Table of n, a(n) for n = 1..992
Wikipedia, Zahlen in unterschiedlichen Sprachen
Wikipedia, List of numbers in various languages

Intersection of A010784 and Cf. A247751.

Cf. A247800 (Czech), A247802 (Dutch), A053433 (English), A247803 (Finnish), A247804 (French), A247805 (German), A247806 (Hungarian), A247807 (Italian), A247808 (Latin), A247809 (Norwegian), A247810 (Polish), A247807 (Portuguese), A247811 (Russian), A247812 (Slovak), A247813 (Spanish), A247809 (Swedish), A247814 (Turkish).

import Data.IntSet (fromList, deleteFindMin, union)
import qualified Data.IntSet as Set (null)
a247801 n = a247801_list !! (n-1)
a247801_list = 0 : f (fromList [1..9]) where
   f s | Set.null s = []
       | otherwise  = x : f (s' `union`
         fromList (map (+ 10 * x) $ tail $ dropWhile (/= mod x 10) digs))
       where (x, s') = deleteFindMin s
   digs = [1, 5, 4, 9, 0, 8, 6, 7, 2, 3]

cp's OEIS Frontend

A178787 Number of numbers <= n having distinct digits in their decimal representation, cf. A010784.

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A321149 a(1) = 102735, a(n) = prime(n-1)*a(n-1) but products that are not in A010784 are first reduced as in A320486 (see comments); continue until zero is reached.

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A321148 a(1) = 24603, a(n) = n*a(n-1) but products that are not in A010784 are first reduced as in A320486 (see comments); continue until zero is reached.

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A160343 Numbers k such that the two closest numbers above and below k, which are in A010784 and which have no common digit with k, have the same distance to k.

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Extensions

A043096 Numbers in which every pair of adjacent digits are distinct.

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Maple

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A029743 Primes with distinct digits.

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A178788 Characteristic function of numbers having distinct digits in their decimal representation.

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A247802 Numbers in decimal representation with distinct digits, such that in Dutch their digits are in alphabetic order.

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