A342441 -id:A342441 - cp's OEIS frontend

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).

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

A362075 a(1) = 1, a(2) = 2; for n > 2, a(n) is the least positive integer not occurring earlier such that a(n) shares no digit with a(n-2) + a(n-1).

Original entry on oeis.org

1, 2, 4, 3, 5, 6, 7, 8, 9, 20, 10, 11, 30, 22, 13, 12, 14, 15, 16, 24, 17, 23, 18, 25, 19, 21, 26, 28, 27, 29, 31, 32, 40, 33, 41, 35, 34, 37, 36, 42, 39, 43, 44, 45, 46, 38, 50, 47, 48, 60, 49, 52, 53, 62, 63, 64, 54, 55, 56, 57, 58, 66, 59, 67, 70, 65, 68, 69, 80, 72, 73, 76, 75, 74, 77, 78

Offset: 1

Scott R. Shannon, Apr 08 2023

The sequence is likely to be finite although it contains at least 100000 terms.

Sequence is finite with 4128755 terms, since a(4128754) = 46946449 and a(4128755) = 777000707 have sum 823947156. - Michael S. Branicky, Apr 08 2023

			a(10) = 20 as a(8) + a(9) = 8 + 9 = 17, and 20 is the smallest unused number that does not contain the digits 1 or 7.

Michael S. Branicky, Table of n, a(n) for n = 1..10000
Michael S. Branicky, Python program
Scott R. Shannon, Image of the first 100000 terms. The green line is a(n) = n.

Cf. A362076, A342441, A342442, A067581, A297065.

# see linked program that generates the full sequence

A342454 a(n) = first prime of the A342444(n) consecutive primes summing to A342443(n).

Original entry on oeis.org

2, 29, 191, 1087, 19979, 34337, 34129, 54829, 1665437, 1428571363, 5882352691, 333333333299, 1550560001, 13384757, 6121296037

Offset: 1

Bernard Schott, Mar 13 2021

			A342439(1) = 2 + 3 = 5 hence a(1) = 2.
A342439(2) = 29 + 31 + 37 = 97 hence a(2) = 29.

Cf. A342440, A342441, A342443, A342444, A342453.

a(5)-a(14) from Jon E. Schoenfield, Mar 14 2021

a(15) from Max Alekseyev, Dec 11 2024

A362093 a(1) = 1, a(2) = 2; for n > 2, a(n) is the least positive integer not occurring earlier such that a(n) shares no digit with a(n-1) - a(n-2).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 20, 22, 10, 30, 11, 23, 33, 24, 12, 34, 13, 35, 14, 36, 15, 37, 16, 38, 17, 39, 18, 40, 19, 43, 31, 44, 25, 26, 27, 28, 29, 32, 21, 42, 45, 41, 50, 46, 51, 47, 52, 48, 53, 49, 55, 54, 56, 57, 58, 59, 60, 62, 61, 63, 64, 65, 66, 67, 68, 69, 70, 72, 71, 73, 74, 75, 76, 77

Offset: 1

Scott R. Shannon, Apr 08 2023

nonn

The sequence is likely to be finite although it contains at least 1 million terms.

Sequence is finite with 6080472 terms, since a(6080471) = 660606060 and a(6080472) = 8822811 have difference -651783249. - Michael S. Branicky, Apr 09 2023

			a(11) = 22 as a(10) - a(9) = 20 - 9 = 11, and 22 is the smallest unused number that does not contain the digit 1.

Michael S. Branicky, Python program
Scott R. Shannon, Image of the first 1 million terms. The green line is a(n) = n.

Cf. A362075, A362076, A342441, A342442, A067581, A297065.

# see link for program that generates full sequence

A362076 a(1) = 1, a(2) = 2; for n > 2, a(n) is the least positive integer not occurring earlier such that a(n) shares no digit with a(n-2) * a(n-1).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 22, 13, 14, 30, 15, 12, 23, 18, 20, 17, 16, 19, 21, 24, 26, 31, 25, 28, 29, 33, 32, 27, 35, 36, 34, 37, 39, 50, 38, 42, 40, 43, 44, 45, 46, 41, 47, 48, 49, 60, 51, 52, 70, 55, 61, 62, 54, 56, 57, 58, 59, 65, 64, 53, 66, 67, 63, 68, 69, 71, 72, 73, 74, 76, 77

Offset: 1

Scott R. Shannon, Apr 08 2023

The sequence is finite; after 6481 terms a(6480) = 5211 and a(6481) = 44444 resulting in a product of 5211 * 44444 = 231597684. This contains all digits 1 to 9 so the next term does not exist.

The sequence contains 40 fixed points, the last being a(5477).

			a(12) = 22 as a(10) * a(11) = 10 * 11 = 110, and 22 is the smallest unused number that does not contain the digits 0 or 1.

Michael S. Branicky, Table of n, a(n) for n = 1..6481
Michael S. Branicky, Python program
Scott R. Shannon, Image of the 6481 terms. The green line is a(n) = n.

Cf. A362075, A342441, A342442, A067581, A297065.

Python
```
# see linked program
```

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

cp's OEIS Frontend

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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A362075 a(1) = 1, a(2) = 2; for n > 2, a(n) is the least positive integer not occurring earlier such that a(n) shares no digit with a(n-2) + a(n-1).

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A342454 a(n) = first prime of the A342444(n) consecutive primes summing to A342443(n).

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A362093 a(1) = 1, a(2) = 2; for n > 2, a(n) is the least positive integer not occurring earlier such that a(n) shares no digit with a(n-1) - a(n-2).

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A362076 a(1) = 1, a(2) = 2; for n > 2, a(n) is the least positive integer not occurring earlier such that a(n) shares no digit with a(n-2) * a(n-1).

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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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