A362075 -id:A362075 - cp's OEIS frontend

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).

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
```

A368181 a(1) = 1; for n > 1, a(n) is the smallest positive integer that has not yet appeared which shares no digit with the sum of all previous terms a(1)..a(n-1).

Original entry on oeis.org

1, 2, 4, 3, 5, 6, 7, 9, 8, 10, 11, 12, 13, 20, 22, 24, 23, 25, 14, 30, 15, 17, 33, 26, 16, 18, 19, 21, 27, 28, 31, 29, 34, 40, 41, 32, 35, 36, 38, 39, 37, 42, 44, 50, 43, 52, 45, 46, 47, 48, 49, 54, 55, 57, 56, 60, 53, 58, 59, 62, 63, 65, 70, 51, 61, 64, 66, 67, 69, 68, 71, 73, 74, 81, 90, 91, 77

Offset: 1

Scott R. Shannon, Dec 21 2023

The sequence is finite; after 14594 terms, where a(14594) = 20858, the sum of all terms is 173658294 which contains the digits 1..9, so the next term does not exist.

The largest term is a(12742) = 888888.

			a(14) = 20 as the sum of all terms a(1)..a(13) = 91, and 20 is the smallest unused number that does not contain the digits 1 or 9.

Scott R. Shannon, Table of n, a(n) for n = 1..14594

Cf. A362093, A362075, A342383, A342382.

from itertools import islice
def agen():
  s, aset, mink = 0, {0}, 1
  while True:
      k, dset = mink, set(str(s))
      if dset >= set("123456789"): break
      while k in aset or set(str(k)) & dset: k += 1
      an = k; aset.add(an); s += an; yield an
      while mink in aset: mink += 1
print(list(islice(agen(), 80))) # Michael S. Branicky, Dec 21 2023

A368347 a(1) = 1; for n > 1, a(n) is the smallest positive integer that has not yet appeared which contains all the distinct digits of the sum of all previous terms a(1)..a(n-1).

Original entry on oeis.org

1, 10, 11, 2, 24, 48, 69, 156, 123, 4, 84, 235, 67, 348, 128, 103, 134, 1457, 304, 308, 136, 2357, 1069, 178, 3567, 10239, 126, 182, 10247, 137, 13458, 12345, 567, 2458, 2068, 20567, 1378, 45689, 10348, 102347, 203479, 4568, 12456, 234568, 105689, 3089, 20689, 12678, 204589, 1048, 1023459

Offset: 1

Scott R. Shannon, Dec 22 2023

The sequence is infinite, although it is unknown if all positive numbers eventually appear. In the first 50000 terms the smallest number not to have appeared is 3. In the same range the largest value is a(49134) = 1023548967, with the sum of all previous terms at that point being 553402987165.

			a(3) = 11 as the sum of the first two terms is 1 + 10 = 11, which contains the distinct digit 1, and 11 is the smallest unused number to contain 1.
a(4) = 2 as the sum of the first three terms is 1 + 10 + 11 = 22, which contains the distinct digit 2, and 2 is the smallest unused number to contain 2.
a(5) = 24 as the sum of the first four terms is 1 + 10 + 11 + 2 = 24, which contains the distinct digits 2 and 4, and 24 is the smallest unused number to contain 2 and 4.

Scott R. Shannon, Table of n, a(n) for n = 1..10000

Cf. A368181, A362093, A362075, A342383, A342382.

A368559 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number not occurring earlier such that a(n) contains all the distinct digits of a(n-1) - a(n-2).

Original entry on oeis.org

1, 2, 10, 8, 12, 4, 18, 14, 24, 100, 67, 3, 46, 34, 21, 13, 28, 15, 31, 16, 51, 35, 61, 26, 53, 27, 62, 135, 37, 89, 25, 64, 39, 52, 103, 105, 20, 58, 38, 102, 146, 40, 106, 6, 101, 59, 42, 17, 125, 108, 71, 73, 22, 115, 93, 23, 70, 47, 32, 145, 113, 123, 104, 19, 85, 36, 49, 130, 81, 94

Offset: 1

Scott R. Shannon, Dec 30 2023

The sequence is conjectured to be a permutation of the positive integers. The fixed points begin 1, 2, 95, 122, 156, 318, 1644, 1964, 2189, 2740, 8264, 16904, ... although it is likely there are infinitely more.

			a(3) = 10 as a(2) - a(1) = 2 - 1 = 1, and 10 is the smallest unused number to contain 1.
a(11) = 67 as a(10) - a(9) = 100 - 24 = 76, and 67 is the smallest unused number to contain 7 and 6.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image of the first 250000 terms. The green line is a(n) = n.

Cf. A362093, A362075, A184992, A069638.

cp's OEIS Frontend

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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A368181 a(1) = 1; for n > 1, a(n) is the smallest positive integer that has not yet appeared which shares no digit with the sum of all previous terms a(1)..a(n-1).

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A368347 a(1) = 1; for n > 1, a(n) is the smallest positive integer that has not yet appeared which contains all the distinct digits of the sum of all previous terms a(1)..a(n-1).

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A368559 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number not occurring earlier such that a(n) contains all the distinct digits of a(n-1) - a(n-2).

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