A342265 -id:A342265 - cp's OEIS frontend

A342264 Lexicographically earliest sequence of distinct nonnegative terms such that both a(n) and a(n) + a(n+1) have digits in nondecreasing order.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 13, 11, 12, 14, 15, 18, 16, 17, 19, 25, 22, 23, 24, 33, 26, 29, 27, 28, 38, 39, 49, 66, 45, 34, 35, 44, 55, 56, 57, 58, 59, 67, 46, 68, 47, 69, 48, 77, 36, 78, 37, 79, 88, 89, 99, 123, 111, 112, 113, 114, 115, 118, 116, 117, 119, 125, 122, 124, 133, 126, 129, 127, 128, 138

Offset: 1

Eric Angelini and Carole Dubois, Mar 07 2021

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

			a(10) = 9 and a(11) = 13 sum up to 22: the three numbers have digits in nondecreasing order;
a(11) = 13 and a(12) = 11 sum up to 24 (same property);
a(12) = 11 and a(13) = 12 sum up to 23 (same property);
etc.

Robert Israel, Table of n, a(n) for n = 1..4000

Cf. A009994 (numbers with digits in nondecreasing order), A342265 and A342266 (variations on the same idea).

ND[1]:= [$1..9]:
for d from 2 to 5 do
   ND[d]:= map(proc(t) local j; seq(10*t + j,j=(t mod 10) .. 9) end proc, ND[d-1])
od:
S:= [seq(op(ND[i]),i=1..5)]): nS:= nops(S):
isnd:= proc(x) member(x,ND[ilog10(x)+1]) end proc:
R:= 0: t:= 0:
for count from 2 to 100 do
  found:= false;
  for i from 1 to nS do
    if isnd(t + S[i]) then
      R:= R, S[i];
      t:= S[i];
      S:= subsop(i=NULL, S);
      nS:= nS-1;
      found:= true;
      break
    fi;
  od;
  if not found then break fi;
od:
R; # Robert Israel, Jul 14 2025

def nondec(n): s = str(n); return s == "".join(sorted(s))
def aupton(terms):
  alst = [0]
  for n in range(2, terms+1):
    an = 1
    while True:
      while an in alst: an += 1
      if nondec(an) and nondec(alst[-1]+an): alst.append(an); break
      else: an += 1
  return alst
print(aupton(74)) # 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

A342264 Lexicographically earliest sequence of distinct nonnegative terms such that both a(n) and a(n) + a(n+1) have digits in nondecreasing order.

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