A010784 -id:A010784 - cp's OEIS frontend

A280594 Nonnegative numbers whose digits can be formed by typing adjacent keys on a 123-456-789-X0X keypad without repeating a digit.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 14, 21, 23, 25, 32, 36, 41, 45, 47, 52, 54, 56, 58, 63, 65, 69, 74, 78, 80, 85, 87, 89, 96, 98, 123, 125, 145, 147, 214, 236, 254, 256, 258, 321, 325, 365, 369, 412, 452, 456, 458, 478, 521, 523, 541, 547, 563, 569, 580, 587, 589, 632, 652, 654, 658, 698, 741

Offset: 1

FUNG Cheok Yin, Jan 06 2017

Number of terms < 10^k for k = 1,2,3,...: 10, 35, 82, 167, 281, 419, 547, 669, 723. - Robert G. Wilson v, Feb 06 2017

A subsequence of A010784. - FUNG Cheok Yin, Jul 05 2018

			The keypad is:
+---+---+---+
| 1 | 2 | 3 |
+---+---+---+
| 4 | 5 | 6 |
+---+---+---+
| 7 | 8 | 9 |
+---+---+---+
| x | 0 | x |
+---+---+---+
It is visibly obvious that 2580 can be formed on the keypad.

FUNG Cheok Yin, Table of n, a(n) for n = 1..723

Cf. A010784, A280593, A280595.

g = Graph[{1 <-> 2, 1 <-> 4,
    2 <-> 1, 2 <-> 3, 2 <-> 5,
    3 <-> 2, 3 <-> 6,
    4 <-> 1, 4 <-> 5, 4 <-> 7,
    5 <-> 2, 5 <-> 4, 5 <-> 6, 5 <-> 8,
    6 <-> 3, 6 <-> 5, 6 <-> 9,
    7 <-> 4, 7 <-> 8,
    8 <-> 0, 8 <-> 5, 8 <-> 7, 8 <-> 9,
    9 <-> 6, 9 <-> 8}];
f[{a_, b_}] := FindPath[g, a, b, Infinity, All]
ff = f /@ Flatten[Outer[List, r = Range[9], Range[0, 9]], 1];
A280594 = Sort[Join[r, FromDigits /@ Flatten[ff, 1]]] (* Jean-François Alcover, Jan 07 2017 *)

Initial 0 prefixed by N. J. A. Sloane, Feb 05 2017

A342441 a(1) = 1; for n > 1, a(n) is the least positive integer not occurring earlier such that a(n-1)+a(n) shares no digit with either a(n-1) or a(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 14, 15, 17, 16, 18, 12, 21, 19, 25, 22, 23, 26, 24, 27, 28, 29, 31, 33, 32, 34, 35, 36, 38, 39, 41, 42, 43, 37, 44, 45, 46, 47, 48, 51, 149, 53, 49, 52, 54, 55, 56, 57, 59, 58, 63, 64, 65, 66, 67, 68, 62, 69, 72, 73, 75, 85, 77, 74, 76, 78, 82, 79

Offset: 1

Scott R. Shannon, Mar 12 2021

No term can end in 0 as that would result in the last digit of a(n-1) being the same as the last digit of a(n-1)+a(n).

			a(2) = 2 as a(1)+2 = 1+2 = 3 which shares no digit with a(1) = 1 or 2.
a(10) = 11 as a(9)+11 = 9+11 = 20 which shares no digit with a(9) = 9 or 11. Note that the first number skipped is 10 as 9+10 = 19 which shares a digit with 9.
a(11) = 13 as a(10)+13 = 11+13 = 24 which shares no digit with a(10) = 11 or 13. Note that the number 12 is skipped as 11+12 = 23 which shares a digit with 12.

Scott R. Shannon, Image of the first 100000 terms. The green line is a(n) = n.

Cf. A342442 (multiplication), A276633, A010784, A043537, A043096, A338466, A336285.

Block[{a = {1}, m = {1}, d, s, k}, Do[k = 2; While[Nand[FreeQ[a, k], ! IntersectingQ[Set[d, IntegerDigits[k]], Set[s, IntegerDigits[a[[-1]] + k]]], ! IntersectingQ[s, m]], k++]; AppendTo[a, k]; Set[m, d], 72]; a] (* Michael De Vlieger, Mar 20 2021 *)

def aupton(terms):
  alst, aset = [1], {1}
  while len(alst) < terms:
    an, anm1_digs = 2, set(str(alst[-1]))
    while True:
      while an in aset: an += 1
      if (set(str(an)) | anm1_digs) & set(str(an+alst[-1])) == set():
        alst.append(an); aset.add(an); break
      an += 1
  return alst
print(aupton(73)) # Michael S. Branicky, Mar 20 2021

A210666 Numbers with at least three digits in which all digits but one are the same.

Original entry on oeis.org

100, 101, 110, 112, 113, 114, 115, 116, 117, 118, 119, 121, 122, 131, 133, 141, 144, 151, 155, 161, 166, 171, 177, 181, 188, 191, 199, 200, 202, 211, 212, 220, 221, 223, 224, 225, 226, 227, 228, 229, 232, 233, 242, 244, 252, 255, 262, 266, 272, 277, 282, 288

Offset: 1

Arkadiusz Wesolowski, May 08 2012

Each k-digit term has k-1 appearances of a digit, d1, and 1 appearance of a different digit, d2, and k-1 >= 2 so that d1 is repeated.  Specifically, the 2-digit terms of A010784 are not terms here. - Michael S. Branicky, May 22 2022
a(n) = A031955(n+81) for n <= 244.
For n <= 243, i.e., the 3-digit terms, a(n) = A218556(n+10). - M. F. Hasler, Nov 02 2012

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000

Subsequence of A031955. Supersequence of A164937.

Cf. A010784, A031955, A218556.

lst = {}; Do[If[SortBy[Tally[IntegerDigits[n]], Last][[-1, -1]] == IntegerLength[n] - 1, AppendTo[lst, n]], {n, 100, 288}]; lst
lst = {}; Do[r = Table[a, {n}]; Do[c = FromDigits@Permutations[Join[{d}, r]]; If[d == 0, c = Rest[c]]; AppendTo[lst, c], {d, 0, 9}], {a, 0, 9}, {n, 2, 2}]; Drop[Union@Flatten[lst], 19]
nrepQ[n_] := Module[{dg = Select[DigitCount[n], # > 0 &]}, Length[dg] == 2 && Min[dg] == 1 && Max[dg] > 1]; Select[Range[300], nrepQ] (* Harvey P. Dale, Nov 20 2012 *)

from itertools import count, islice
def agen(): # generator of terms
    for d in count(3):
        dterms = set()
        for most in "123456789":
            dterms.add(int(most + "0"*(d-1)))
            for diff in "0123456789":
                if most == diff: continue
                cands = (most*i + diff + most*(d-1-i) for i in range(d))
                dterms.update(int(t) for t in cands if t[0] != "0")
        yield from sorted(dterms)
print(list(islice(agen(), 52))) # Michael S. Branicky, May 17 2022

A280595 Nonnegative numbers whose digits can be formed by typing adjacent keys on a 123-456-789-00X keypad without repeating a digit.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 14, 21, 23, 25, 32, 36, 41, 45, 47, 52, 54, 56, 58, 63, 65, 69, 70, 74, 78, 80, 85, 87, 89, 96, 98, 123, 125, 145, 147, 214, 236, 254, 256, 258, 321, 325, 365, 369, 412, 452, 456, 458, 470, 478, 521, 523, 541, 547, 563, 569, 580, 587, 589, 632, 652, 654, 658

Offset: 1

FUNG Cheok Yin, Jan 06 2017

A subsequence of A010784. - FUNG Cheok Yin, Jul 05 2018

			The keypad is:
+---+---+---+
| 1 | 2 | 3 |
+---+---+---+
| 4 | 5 | 6 |
+---+---+---+
| 7 | 8 | 9 |
+---+---+---+
|   0   | x |
+---+---+---+
It is visibly obvious that 5807 can be formed on the keypad.

FUNG Cheok Yin, Table of n, a(n) for n = 1..1082

Cf. A010784, A280593, A280594.

g = Graph[{1 <-> 2, 1 <-> 4,
    2 <-> 1, 2 <-> 3, 2 <-> 5,
    3 <-> 2, 3 <-> 6,
    4 <-> 1, 4 <-> 5, 4 <-> 7,
    5 <-> 2, 5 <-> 4, 5 <-> 6, 5 <-> 8,
    6 <-> 3, 6 <-> 5, 6 <-> 9,
    7 <-> 0, 7 <-> 4, 7 <-> 8,
    8 <-> 0, 8 <-> 5, 8 <-> 7, 8 <-> 9,
    9 <-> 6, 9 <-> 8}];
f[{a_, b_}] := FindPath[g, a, b, Infinity, All]
ff = f /@ Flatten[Outer[List, r = Range[9], Range[0, 9]], 1];
A280595 = Sort[Join[r, FromDigits /@ Flatten[ff, 1]]] (* Jean-François Alcover, Jan 07 2017 *)

Initial 0 added by N. J. A. Sloane, Feb 05 2017

A336285 a(0) = 0; for n > 0, a(n) is the least positive integer not occurring earlier such that the digits in a(n-1)+a(n) are all distinct.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 53, 50, 52, 51, 54, 55, 65, 58, 62, 61, 59, 64, 56, 67, 57, 63, 60, 66, 68, 69, 70, 72, 71, 74, 73, 75, 77

Offset: 0

Rémy Sigrist, Jul 22 2020

In other words, for any n > 0, a(n) + a(n+1) belongs to A010784.

The sequence is finite since there are only a finite number of positive integers with distinct digits, see A010784, although the exact number of terms is currently unknown.

			The first terms, alongside a(n) + a(n+1), are:
  n   a(n)  a(n)+a(n+1)
  --  ----  -----------
   0     0            1
   1     1            3
   2     2            5
   3     3            7
   4     4            9
   5     5           12
   6     7           13
   7     6           14
   8     8           17
   9     9           19
  10    10           21

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Scott R. Shannon, Image of the first 1000000 terms. The green line is a(n) = n.

Cf. A342383, A338466, A322845, A010784, A043537, A043096, A276633, A002378.

s=0; v=1; for (n=1, 67, print1 (v", "); s+=2^v; for (w=1, oo, if (!bittest(s, w) && #(d=digits(v+w))==#Set(d), v=w; break)))

def agen():
  alst, aset, min_unused = [0], {0}, 1
  yield 0
  while True:
    an = min_unused
    while True:
      while an in aset: an += 1
      t = str(alst[-1] + an)
      if len(t) == len(set(t)):
        alst.append(an); aset.add(an); yield an
        if an == min_unused: min_unused = min(set(range(max(aset)+2))-aset)
        break
      an += 1
g = agen()
print([next(g) for n in range(77)]) # Michael S. Branicky, Mar 11 2021

a(0)=0 added by N. J. A. Sloane, Mar 14 2021

A338466 a(0) = 0; for n > 0, a(n) is the least positive integer not occurring earlier such that the digits in a(n-1)*a(n) are all distinct.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 11, 13, 14, 15, 16, 19, 18, 17, 20, 21, 22, 23, 26, 24, 27, 25, 29, 28, 30, 31, 33, 32, 39, 34, 37, 35, 36, 38, 40, 41, 43, 42, 45, 44, 47, 50, 49, 52, 48, 55, 46, 51, 53, 56, 54, 57, 60, 58, 62, 59, 66, 61, 64, 63, 65, 71, 70, 67, 69, 68, 72, 74, 73, 77, 79

Offset: 0

Scott R. Shannon, Mar 09 2021

The sequence is finite, the 71782nd term being a(71781) = 50005 beyond which no number exists that has not occurred earlier such that 50005*a(n) has distinct digits. The maximum term is a(71428) = 175446.

			a(1) = 1 as a(0)*1 = 0*1 = 0 which has one distinct digit 0.
a(10) = 10 as a(9)*10 = 9*10 = 90 which has two distinct digits 9 and 0.
a(11) = 12 as a(10)*12 = 10*12 = 120 which has three distinct digits. Note that 11 is the first skipped number as 10*11 = 110 which has 1 as a duplicate digit.
a(12) = 11 as a(11)*11 = 12*11 = 132 which has three distinct digits.

Scott R. Shannon, Image of the 71782 terms. The green line is a(n) = n.

Cf. A342382, A010784, A043537, A043096, A276633, A002378.

Offset corrected by N. J. A. Sloane, Jun 16 2021

A342442 a(1) = 2; for n > 1, a(n) is the least positive integer not occurring earlier such that a(n-1)*a(n) shares no digit with either a(n-1) or a(n).

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 42, 14, 17, 18, 15, 16, 13, 19, 32, 22, 23, 26, 29, 12, 25, 24, 34, 27, 33, 36, 39, 43, 37, 38, 28, 47, 44, 45, 46, 63, 66, 65, 48, 49, 62, 55, 54, 35, 174, 53, 76, 57, 56, 59, 52, 58, 64, 92, 74, 68, 78, 72, 77, 67, 73, 83, 69, 79, 84, 75, 88, 113, 183, 138, 149, 148

Offset: 1

Scott R. Shannon, Mar 12 2021

No term can end in 0 or 1 as that would result in the last digit of a(n-1)*a(n) being the same as a(n)'s last digit. The majority of terms appear to grow linearly with n but occasional large spikes in the values also occur, e.g. a(47888) = 425956849. See the examples. It is unknown if the sequence is infinite.

			a(2) = 3 as a(1)*3 = 2*3 = 6 which shares no digit with a(1) = 2 or 3.
a(9) = 42 as a(8)*42 = 9*42 = 378 which shares no digit with a(8) = 9 or 42.
a(10) = 14 as a(9)*14 = 42*14 = 588 which shares no digit with a(9) = 42 or 14.
a(47888) = 425956849 as a(47887)*425956849 = 258649*425956849 = 110173313037001 which shares no digit with a(47887) = 258649 or 425956849.

Daniel Mondot, Table of n, a(n) for n = 1..10000

Cf. A342441 (addition), A342755, A276633, A010784, A043537, A043096, A338466, A336285.

def aupton(terms):
  alst, aset = [2], {2}
  while len(alst) < terms:
    an, anm1_digs = 2, set(str(alst[-1]))
    while True:
      while an in aset: an += 1
      if (set(str(an)) | anm1_digs) & set(str(an*alst[-1])) == set():
        alst.append(an); aset.add(an); break
      an += 1
  return alst
print(aupton(74)) # Michael S. Branicky, Mar 20 2021

A156071 Concatenation chain arising in A156069.

Original entry on oeis.org

3, 38, 381, 3816, 38165, 381654, 3816547, 38165472, 381654729

Offset: 1

Alexander R. Povolotsky, Sep 25 2009

a(9) is a zeroless pandigital number in base 10, with 9 digits such that every k-digit substring ( 1 <= k <= 9 ) taken from the left, is divisible by k (see A163574). - Michel Marcus, Dec 01 2013

Matt Parker, Things to make and do in the fourth dimension, 2015, pages 7-9.

Blaine, How about a math puzzle?
Albert Franck, Puzzles, see item 7.
Wikipedia, Polydivisible number
Jim Wilson, Statement of Nine-Digit Number puzzle

Cf. A010784, A030299, A050289, A143671, A144688, A156069, A158242, A163574.

A342382 a(0) = 0; for n > 0, a(n) is the least positive integer not occurring earlier such that both the digits in a(n) and the digits in a(n-1)*a(n) are all distinct.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 19, 18, 17, 20, 21, 23, 26, 24, 27, 25, 29, 28, 30, 31, 34, 37, 35, 36, 38, 39, 32, 40, 41, 43, 42, 45, 48, 52, 49, 50, 47, 51, 46, 53, 56, 54, 57, 60, 58, 62, 59, 68, 64, 61, 65, 63, 72, 69, 67, 70, 71, 73, 74, 76, 78, 80, 79, 82, 75, 81, 83

Offset: 0

Scott R. Shannon, Mar 09 2021

The sequence is finite, the 18351st term being a(18350) = 41987 beyond which no number exists that has not occurred earlier that has all distinct digits and that 41987*a(n) has all distinct digits. The maximum term is a(18097) = 219087.

			a(1) = 1 as 1 has one distinct digit and a(0)*1 = 0*1 = 0 which has one distinct digit 0.
a(10) = 10 as 10 has two distinct digits and a(9)*10 = 9*10 = 90 which has two distinct digits 9 and 0.
a(11) = 12 as 12 has two distinct digits and a(10)*12 = 10*12 = 120 which has three distinct digits. Note that 11 is the first skipped number as 11 has 1 as a duplicate digit.
a(16) = 19 as 19 has two distinct digits and a(15)*19 = 16*19 = 304 which has three distinct digits. Note that 17 and 18 are skipped as 16*17 = 272 while 16*18 = 288, both of which contain duplicate digits.

Scott R. Shannon, Image of the 18351 terms. The green line is a(n) = n.

Cf. A338466, A010784, A043537, A043096, A276633, A002378.

Block[{a = {0}, k, m = 42000}, Do[k = 1; While[Nand[FreeQ[a, k], AllTrue[DigitCount[a[[-1]]*k], # < 2 &], AllTrue[DigitCount[k], # < 2 &]], If[k > m, Break[]]; k++]; If[k > m, Break[]]; AppendTo[a, k], {i, 76}]; a] (* Michael De Vlieger, Mar 11 2021 *)

Offset corrected by N. J. A. Sloane, Jun 16 2021

A029740 Odd numbers with distinct digits.

Original entry on oeis.org

1, 3, 5, 7, 9, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 103, 105, 107, 109, 123, 125, 127, 129, 135, 137, 139, 143, 145, 147, 149, 153, 157

Offset: 1

Patrick De Geest

The sequence has 4384045 terms. - Harvey P. Dale, Jan 12 2019

Maximum term is 9876543201. Next higher number with distinct digits is 9876543210. - Alonso del Arte, Jan 09 2020

Michael S. Branicky, Table of n, a(n) for n = 1..10000
Patrick De Geest, World!Of Numbers

Cf. A029741 (even version). Union of that sequence with this sequence gives A010784.

Select[Range[1, 199, 2], Max[DigitCount[#]] == 1 &] (* Harvey P. Dale, Jan 12 2019 *)

# generates full sequence
from itertools import permutations
afull = sorted(set(int("".join(p)) for d in range(1, 11) for p in permutations("0123456789", d) if p[0] != "0" and p[-1] in "13579"))
print(afull[:100]) # Michael S. Branicky, Aug 04 2022

def hasDistinctDigits(n: Int): Boolean = {
  val numerStr = n.toString
  val digitSet = numerStr.split("").toSet
  numerStr.length == digitSet.size
}
(1 to 199 by 2).filter(hasDistinctDigits) // Alonso del Arte, Jan 09 2020

First comment corrected by Harvey P. Dale, Mar 04 2020 at the insistence of Sean A. Irvine
Offset changed to 1 by Michael S. Branicky, Aug 04 2022
Removed incorrect Sage program. - N. J. A. Sloane, Aug 04 2022

cp's OEIS Frontend

A280594 Nonnegative numbers whose digits can be formed by typing adjacent keys on a 123-456-789-X0X keypad without repeating a digit.

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A342441 a(1) = 1; for n > 1, a(n) is the least positive integer not occurring earlier such that a(n-1)+a(n) shares no digit with either a(n-1) or a(n).

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A210666 Numbers with at least three digits in which all digits but one are the same.

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A280595 Nonnegative numbers whose digits can be formed by typing adjacent keys on a 123-456-789-00X keypad without repeating a digit.

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A336285 a(0) = 0; for n > 0, a(n) is the least positive integer not occurring earlier such that the digits in a(n-1)+a(n) are all distinct.

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A338466 a(0) = 0; for n > 0, a(n) is the least positive integer not occurring earlier such that the digits in a(n-1)*a(n) are all distinct.

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A342442 a(1) = 2; for n > 1, a(n) is the least positive integer not occurring earlier such that a(n-1)*a(n) shares no digit with either a(n-1) or a(n).

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A156071 Concatenation chain arising in A156069.

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