A342264 -id:A342264 - cp's OEIS frontend

A342265 Lexicographically earliest sequence of distinct nonnegative terms such that both a(n) and the cumulative sum a(1)+a(2)+...+a(n) have digits in nondecreasing order.

0, 1, 2, 3, 5, 4, 7, 6, 8, 9, 11, 12, 44, 13, 14, 16, 22, 45, 15, 18, 23, 55, 24, 88, 33, 77, 34, 78, 111, 333, 17, 19, 79, 29, 89, 25, 99, 199, 112, 444, 26, 28, 56, 35, 188, 113, 119, 556, 114, 999, 122, 1199, 888, 123, 4444, 36, 66, 124, 118, 222, 445, 115, 129, 67, 133, 667, 134, 68, 889, 223

Offset: 1

Eric Angelini and Carole Dubois, Mar 07 2021

10 is obviously the first integer not present in the sequence as 1 > 0.

The last term is a(173) = 122222, at which point the cumulative sum is 12467999, and the sequence cannot be extended. - Michael S. Branicky, Feb 05 2024

			Terms a(1) = 0 to a(5) = 5 sum up to 11: those six numbers have digits in nondecreasing order;
terms a(1) = 0 to a(6) = 4 sum up to 15: those seven numbers have digits in nondecreasing order;
terms a(1) = 0 to a(7) = 7 sum up to 22: those eight numbers have digits in nondecreasing order; etc.

Michael S. Branicky, Table of n, a(n) for n = 1..173

Cf. A009994 (numbers with digits in nondecreasing order), A342264 and A342266 (variations on the same idea).

def nondec(n): s = str(n); return s == "".join(sorted(s))
def aupton(terms):
  alst = [0]
  for n in range(2, terms+1):
    an, cumsum = 1, sum(alst)
    while True:
      while an in alst: an += 1
      if nondec(an) and nondec(cumsum + an): alst.append(an); break
      else: an += 1
  return alst
print(aupton(100)) # Michael S. Branicky, Mar 07 2021

A342266 Lexicographically earliest sequence of distinct nonnegative terms such that both a(n) and a(n) * a(n+1) have digits in nondecreasing order.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 8, 7, 5, 9, 13, 12, 14, 16, 18, 26, 44, 27, 17, 15, 23, 29, 46, 28, 48, 47, 24, 19, 117, 38, 36, 33, 34, 37, 67, 35, 127, 114, 39, 57, 78, 146, 236, 58, 77, 1444, 177, 157, 2477, 144, 247, 45, 25, 49, 227, 147, 257, 1777, 12568, 116, 68, 66, 118, 113, 59, 226, 148, 166, 134, 167, 334

Offset: 1

Eric Angelini and Carole Dubois, Mar 07 2021

10 is obviously the first integer not present in the sequence as 1 > 0; 11 will never show either because the result of a(n) * 11 is already in the sequence or because the said result has digits in contradiction with the definition.
It would be good to have a proof that there is an infinite sequence with the desired property. It could happen then any choice for any number of initial terms will eventually fail. - David A. Corneth and N. J. A. Sloane, Mar 07 2021
The authors agree, but are unable to give the desired proof. So it is indeed possible that this sequence is wrong from the first term on.

			a(5) = 4 and a(6) = 6 have product 24: the three numbers have digits in nondecreasing order;
a(6) = 6 and a(7) = 8 have product 48: the three numbers have digits in nondecreasing order;
a(7) = 8 and a(7) = 7 have product 56: the three numbers have digits in nondecreasing order; etc.

Carole Dubois, Table of n, a(n) for n = 1..138 (terms calculated with a(n+1) < a(n)*1000, (backtracking when not found inside this limit)).

Cf. A009994 (numbers with digits in nondecreasing order), A342264 and A342265 (variations on the same idea).

cp's OEIS Frontend

A342265 Lexicographically earliest sequence of distinct nonnegative terms such that both a(n) and the cumulative sum a(1)+a(2)+...+a(n) have digits in nondecreasing order.

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