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

%I A329299 #11 Oct 14 2022 12:21:00
%S A329299 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,
%T A329299 111,112,113,114,115,116,122,123,124,125,133,134,188,222,223,224,233,
%U A329299 277,278,366,367,368,377,455,456,457,458,466,467,556,557,566,1113
%N A329299 Numbers whose digits are in nondecreasing order in bases 9 and 10.
%C A329299 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?
%e A329299 Sequence includes, respectively, 9, 16, 32, and 11 terms that are 1-, 2-, 3-, and 4- digit terms in both bases, and the following:
%e A329299   a(69) =    14777 =    22238_9
%e A329299   a(70) =    15677 =    23448_9
%e A329299   a(71) =    22234 =    33444_9
%e A329299   a(72) =    22235 =    33445_9
%e A329299   a(73) =    22236 =    33446_9
%e A329299   a(74) =    22237 =    33447_9
%e A329299   a(75) =    22238 =    33448_9
%e A329299   a(76) =    22244 =    33455_9
%e A329299   a(77) =    22245 =    33456_9
%e A329299   a(78) =    22246 =    33457_9
%e A329299   a(79) =    22247 =    33458_9
%e A329299   a(80) =    22255 =    33467_9
%e A329299   a(81) =    22256 =    33468_9
%e A329299   a(82) =    22335 =    33566_9
%e A329299   a(83) =    22336 =    33567_9
%e A329299   a(84) =    22337 =    33568_9
%e A329299   a(85) =    22345 =    33577_9
%e A329299   a(86) =    22346 =    33578_9
%e A329299   a(87) =    22355 =    33588_9
%e A329299   a(88) =    44468 =    66888_9
%e A329299   a(89) =   222344 =   367888_9
%e A329299   a(90) =  1233467 =  2278888_9
%e A329299   a(91) = 12555566 = 25555888_9
%p A329299 filter:= proc(n) local L;
%p A329299   L:= convert(n,base,10);
%p A329299   `and`(seq(L[i+1]<=L[i],i=1..nops(L)-1))
%p A329299 end proc:
%p A329299 ND[1]:= [$1..8]: R:= $0..8:
%p A329299 for d from 2 to 10 do
%p A329299   ND[d]:= map(t -> seq(9*t+r, r=(t mod 9) ..8), ND[d-1]);
%p A329299   R:= R, op(select(filter, ND[d]));
%p A329299 od:
%p A329299 R; # _Robert Israel_, Nov 20 2019
%t A329299 Select[Range[0,1200],Min[Differences[IntegerDigits[#]]]>-1&& Min[ Differences[ IntegerDigits[ #,9]]]>-1&] (* _Harvey P. Dale_, Oct 14 2022 *)
%Y A329299 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.
%K A329299 nonn,base
%O A329299 1,3
%A A329299 _Jon E. Schoenfield_, Nov 17 2019