A336285 -id:A336285 - cp's OEIS frontend

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).

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

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

A342383 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, 7, 6, 8, 9, 10, 13, 12, 14, 15, 16, 18, 17, 19, 20, 21, 24, 23, 25, 26, 27, 29, 28, 30, 31, 32, 35, 34, 36, 37, 38, 40, 39, 41, 42, 43, 46, 45, 47, 48, 49, 53, 50, 52, 51, 54, 69, 56, 64, 59, 61, 62, 58, 65, 60, 63, 57, 67, 68, 70, 72, 71, 74, 73, 75, 78, 76, 80, 79, 81, 82, 83

Offset: 0

Scott R. Shannon, Mar 09 2021

The sequence is finite due to the finite number of positive integers with distinct digits, see A010784, although the exact number of terms is currently unknown.

			a(1) = 1 as 1 has one distinct digit and a(0)+1 = 0+1 = 1 which has one distinct digit 0.
a(6) = 7 as 7 has one distinct digit and a(5)+7 = 5+7 = 12 which has two distinct digits. Note that 6 is the first skipped number as a(5)+6 = 5+6 = 11 has 1 as a duplicate digit.
a(11) = 13 as 13 has two distinct digits and a(10)+13 = 10+13 = 23 which has two distinct digits. Note that 11 and 12 are skipped as 11 has 1 as a duplicate digit while a(10)+12 = 10+12 = 22 has 2 as a duplicate digit.

Scott R. Shannon, Image of the first 60000 terms. The green line is a(n) = n.

Cf. A336285, A342382, A010784, A043537, A043096, A276633, A002378.

Block[{a = {0}, k, m = 10^4}, 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 *)

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

A342755 a(1) = 2; for n > 1, a(n) is the least positive integer not occurring earlier such that a(n) shares no digit with a(n-1) and 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, 15, 22, 14, 55, 12, 37, 16, 25, 36, 29, 47, 23, 46, 13, 44, 18, 32, 17, 38, 19, 33, 26, 35, 174, 53, 76, 59, 34, 27, 43, 67, 49, 62, 87, 106, 493, 57, 24, 75, 48, 65, 122, 39, 54, 72, 88, 45, 66, 73, 56, 77, 52, 79, 84, 63, 78, 123, 69, 58, 64, 92, 74, 68, 114, 85, 314

Offset: 1

Scott R. Shannon, Mar 20 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. Currently the last known term is a(173) = 922989, the next being at least 5*10^10 if it exists. It is unknown if the sequence is infinite.

a(174) = 60060666070067077 and a(175) has 52 digits (see b-file). If a(176) exists, it is > 10^71. - Michael S. Branicky, Apr 10 2023

			a(2) = 3 as 3 shares no digit with a(1) = 2 and a(1)*3 = 2*3 = 6 shares no digit with a(1) = 2 or 3.
a(9) = 42 as 42 shares no digit with a(8) = 9 and a(8)*42 = 9*42 = 378 shares no digit with a(8) = 9 or 42.
a(10) = 15 as 15 shares no digit with a(9) = 42 and a(9)*15 = 42*15 = 630 shares no digit with a(9) = 42 or 15. This is the first term that differs from A342442.
a(173) = 922989 as 922989 shares no digit with a(172) = 7154 and a(172)*922989 = 7154*922989 = 6603063306 shares no digit with a(172) = 7154 or 922989. This is currently the last known term.

Michael S. Branicky, Table of n, a(n) for n = 1..175

Cf. A342442, A342441, 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
      an_digs = set(str(an))
      if (an_digs & anm1_digs) == set():
        prod_digs = set(str(an*alst[-1]))
        if (anm1_digs | an_digs) & prod_digs  == set():
          alst.append(an); aset.add(an); break
      an += 1
  return alst
print(aupton(173)) # Michael S. Branicky, Mar 21 2021

A336478 Lexicographically earliest sequence of distinct positive terms such that the sum of two consecutive terms has distinct digits in primorial base.

Original entry on oeis.org

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

Offset: 1

Rémy Sigrist, Jul 22 2020

In other words, for any n > 0, a(n) + a(n+1) belongs to A321683.

			The first terms, alongside the primorial representation of a(n)+a(n+1), are:
  n   a(n)  prim(a(n)+a(n+1))
  --  ----  -----------------
   1     1    (2,0)
   2     3    (2,1)
   3     2  (1,2,0)
   4     8  (2,0,1)
   5     5  (2,1,0)
   6     9  (2,0,1)
   7     4  (1,2,0)
   8     6  (2,0,1)
   9     7  (3,0,1)
  10    12  (3,2,0)
  11    10  (3,2,1)
  12    13  (4,2,0)
  13    15  (4,1,0)
  14    11  (4,0,1)

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Scatterplot of the first 1000000 terms
Rémy Sigrist, PARI program for A336478
Index entries for sequences related to primorial base

Cf. A321683, A322845, A336285.

PARI
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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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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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A342383 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.

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A342755 a(1) = 2; for n > 1, a(n) is the least positive integer not occurring earlier such that a(n) shares no digit with a(n-1) and a(n-1)*a(n) shares no digit with either a(n-1) or a(n).

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A336478 Lexicographically earliest sequence of distinct positive terms such that the sum of two consecutive terms has distinct digits in primorial base.

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PARI