A329299 Numbers whose digits are in nondecreasing order in bases 9 and 10.

0, 1, 2, 3, 4, 5, 6, 7, 8, 11, 12, 13, 14, 15, 16, 17, 22, 23, 24, 25, 26, 33, 34, 35, 44, 111, 112, 113, 114, 115, 116, 122, 123, 124, 125, 133, 134, 188, 222, 223, 224, 233, 277, 278, 366, 367, 368, 377, 455, 456, 457, 458, 466, 467, 556, 557, 566, 1113

Offset: 1

Author

Jon E. Schoenfield, Nov 17 2019

Keywords

• nonn
• base

Comments

a(91) = 12555566 is the largest term < 10^10000 (which is a 10480-digit number in base 9). But can it be proved that 12555566 is the final term of the sequence?

Examples

			Sequence includes, respectively, 9, 16, 32, and 11 terms that are 1-, 2-, 3-, and 4- digit terms in both bases, and the following:
  a(69) =    14777 =    22238_9
  a(70) =    15677 =    23448_9
  a(71) =    22234 =    33444_9
  a(72) =    22235 =    33445_9
  a(73) =    22236 =    33446_9
  a(74) =    22237 =    33447_9
  a(75) =    22238 =    33448_9
  a(76) =    22244 =    33455_9
  a(77) =    22245 =    33456_9
  a(78) =    22246 =    33457_9
  a(79) =    22247 =    33458_9
  a(80) =    22255 =    33467_9
  a(81) =    22256 =    33468_9
  a(82) =    22335 =    33566_9
  a(83) =    22336 =    33567_9
  a(84) =    22337 =    33568_9
  a(85) =    22345 =    33577_9
  a(86) =    22346 =    33578_9
  a(87) =    22355 =    33588_9
  a(88) =    44468 =    66888_9
  a(89) =   222344 =   367888_9
  a(90) =  1233467 =  2278888_9
  a(91) = 12555566 = 25555888_9
		

Crossrefs

Intersection of A023751 (base 9) and A009994 (base 10). Numbers whose digits are in nondecreasing order in bases b and b+1: A329294 (b=4), A329295 (b=5), A329296 (b=6), A329297 (b=7), A329299 (b=8), this sequence (b=9). See A329300 for the (apparently) largest term of each of these sequences.

Programs

Maple

filter:= proc(n) local L;
  L:= convert(n,base,10);
  `and`(seq(L[i+1]<=L[i],i=1..nops(L)-1))
end proc:
ND[1]:= [$1..8]: R:= $0..8:
for d from 2 to 10 do
  ND[d]:= map(t -> seq(9*t+r, r=(t mod 9) ..8), ND[d-1]);
  R:= R, op(select(filter, ND[d]));
od:
R; # Robert Israel, Nov 20 2019

Mathematica

Select[Range[0,1200],Min[Differences[IntegerDigits[#]]]>-1&& Min[ Differences[ IntegerDigits[ #,9]]]>-1&] (* Harvey P. Dale, Oct 14 2022 *)