A286890 -id:A286890 - cp's OEIS frontend

A300907 a(n) is the least positive integer not yet in the sequence in which the largest digit of a(n-2) appears among its digits; a(1)=1, a(2)=2.

1, 2, 10, 12, 11, 20, 13, 21, 3, 22, 23, 24, 30, 4, 31, 14, 32, 34, 33, 40, 35, 41, 5, 42, 15, 43, 25, 44, 45, 46, 50, 6, 51, 16, 52, 26, 53, 36, 54, 56, 55, 60, 57, 61, 7, 62, 17, 63, 27, 64, 37, 65, 47, 66, 67, 68, 70, 8, 71, 18, 72, 28, 73, 38, 74, 48, 75, 58, 76, 78, 77, 80

Offset: 1

Enrique Navarrete, Mar 14 2018

Starting from the term a(89)=89, every term must contain a 9.

			For n=5, a(n-2) = 10 which has largest digit 1.  The positive integers containing 1 are 1, 10, 11, 12, 13, ... (A011531).  Since 1 and 10 are already in the sequence, a(5) = 11. - _Michael B. Porter_, Mar 17 2018

Cf. A286890.

FromDigits /@ Nest[Function[a, Append[a, Block[{k = 3, d}, While[Nand[FreeQ[a, #], MemberQ[#, Max@ a[[-2]]]] &@ Set[d, IntegerDigits@ k], k++]; d]]], {{1}, {2}}, 70] (* Michael De Vlieger, Mar 16 2018 *)

A303294 a(n) is the least positive integer not yet in the sequence which shares a digit with either a(n-3) or a(n-2) (or with both), but shares no digit with a(n-1); a(1)=0, a(2)=1, a(3)=2.

Original entry on oeis.org

0, 1, 2, 10, 22, 11, 20, 13, 24, 3, 4, 12, 30, 14, 23, 15, 26, 5, 6, 21, 35, 16, 25, 17, 28, 7, 8, 27, 18, 29, 31, 9, 32, 19, 33, 41, 36, 40, 37, 42, 38, 44, 39, 45, 63, 47, 50, 34, 51, 43, 52, 46, 53, 48, 55, 49, 56, 74, 58, 60, 54, 61, 57, 62, 59

Offset: 1

Enrique Navarrete, Apr 20 2018

Apparently there exist only 5 pairs of consecutive integers belonging this sequence, a(k+1)-a(k)=1 for k in (1,2,10,18,26). Respectively those pairs are: (0;1), (1;2), (3;4), (5;6), and (7;8).

It seems that a(j)=j only for j in (12,14,31,53,55,60,71,73,75,82,84,95,102). - R. J. Cano, Apr 22 2018

			a(10)=3 since it shares a digit (3) with a(8)=13, and shares no digit with a(9)=24.

R. J. Cano, Table of n, a(n) for n = 1..10000
R. J. Cano, A sequencer program in PARI.

Cf. A286890, A300907, A302566, A302388.

PARI
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See Cano link.
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A302388 a(n) is the least positive integer not yet in the sequence in which the largest digit of a(n-3) appears among its digits; a(1)=1, a(2)=2, a(3)=3.

Original entry on oeis.org

1, 2, 3, 10, 12, 13, 11, 20, 23, 14, 21, 30, 4, 22, 31, 24, 25, 32, 34, 5, 33, 40, 15, 35, 41, 45, 50, 42, 51, 52, 43, 53, 54, 44, 55, 56, 46, 57, 6, 16, 7, 26, 36, 17, 60, 61, 27, 62, 63, 37, 64, 65, 47, 66, 67, 70, 68, 71, 72, 8, 73, 74, 18, 75, 76, 28, 77, 78, 38, 79, 48, 58, 9, 80

Offset: 1

Enrique Navarrete, Apr 06 2018

The only fixed points are 1,2,3,99.
Starting from a(87)=89, every term must contain a 9.
First differences are bounded by -64 and 71.

			a(7)=11 since the largest digit of a(4)=10 is 1, and 11 is the least positive integer at n=7 that contains 1.

Cf. A286890, A300907.

Nest[Append[#, Block[{k = 4, d}, While[Nand[FreeQ[#[[All, 1]], k], MemberQ[Set[d, IntegerDigits[k]], Max[#[[-3, -1]] ] ] ], k++]; {k, d}]] &, Transpose@ {#, IntegerDigits@ #} &@ Range[3], 71][[All, 1]] (* Michael De Vlieger, Apr 12 2018 *)

firstTerms(m)={my(Seq:list=List([1,2,3]),z,cp,r,ok);cp=vector(10,u,u-1);for(i=4,m,z=vecmax(digits(Seq[i-3]));for(t=1,oo,forvec(y=vector(t,u,[1,#cp]),ok=0;for(j=1,t,if(cp[y[j]]==z,ok=1;break));if(ok,r=fromdigits(vector(t,u,cp[y[u]]));for(w=1,#Seq,if(r==Seq[w],ok=0;break));if(ok,listput(Seq,r);break(2))))));return(Seq)} \\ R. J. Cano, Apr 14 2018

A323708 a(n) is the smallest positive number not yet in the sequence that contains both the smallest and largest digits from a(n-1); a(1)=1.

Original entry on oeis.org

1, 10, 100, 101, 102, 20, 120, 200, 201, 202, 203, 30, 103, 130, 230, 300, 301, 302, 303, 304, 40, 104, 140, 204, 240, 340, 400, 401, 402, 403, 404, 405, 50, 105, 150, 205, 250, 305, 350, 450, 500, 501, 502, 503, 504, 505, 506, 60, 106, 160, 206, 260, 306, 360

Offset: 1

Enrique Navarrete, Jan 24 2019

All terms starting with a(2)=10 must contain the digit 0.
From a(110)=809 onwards all terms must contain the digits 0 and 9.
Note that A011540 can also be defined as the sequence where a(n) is the smallest number not yet in the sequence that contains the smallest digit from a(n-1). See crossrefs.

			a(2)=10 since 10 is the smallest positive number not yet in the sequence that contains the smallest and largest digit (i.e., 1) from a(1)=1.
a(6)=20 since 20 is the smallest positive number not yet in the sequence that contains the smallest and largest digits from a(5)=102.

Cf. A107353, A011540 (smallest digit only), A286890 (largest digit only), A303605.

N:= 1000: # for terms before the first term > N
A[1]:= 1:
S:= [$2..N]:
dmin:= 1: dmax:= 1:
found:= true:
for n from 2 while found do
    found := false;
    for i from 1 to nops(S) do
      L:= convert(convert(S[i],base,10),set);
      if {dmin,dmax} subset L then
         A[n]:= S[i];
         dmax:= max(L);
         dmin:= min(L);
         found:= true;
         S:= subsop(i=NULL, S);
         break
      fi
    od;
od:
convert(A,list); # Robert Israel, Mar 28 2019

Nest[Append[#, Block[{k = 2, d}, While[Nand[FreeQ[#[[All, 1]], k], SubsetQ[Set[d, IntegerDigits[k]], #[[-1, -1]] ]], k++]; {k, {Min@ d, Max@ d}}]] &, {{1, {1, 1}}}, 53][[All, 1]] (* Michael De Vlieger, Jan 27 2019 *)

getFirstTerms(n)={my(Z=List(),A=List([1]),dd=[0,1],c,m=1);for(k=2,+oo,forvec(y=vector(k,u,[u==1,9]),listput(Z,y);for(i=1,#Z,if(m==n,return(Vec(A)));c=2;for(q=1,2,for(j=1,#Z[i],if(Z[i][j]==dd[q],c--;break)));if(!c,dd[1]=vecmin(Z[i]);dd[2]=vecmax(Z[i]);listput(A,fromdigits(Z[i]));listpop(Z,i);m++;break)),0))} \\ R. J. Cano, Feb 04 2019

isok(k, vas, dm, dM) = {if (vecsearch(vas, k), return (0)); my(dk = Set(digits(k))); vecsearch(dk, dm) && vecsearch(dk, dM);}
nexta(va, vas, i) = {my(k=1, d=digits(va[i]), dm = vecmin(d), dM = vecmax(d)); while (!isok(k, vas, dm, dM), k++); k;}
lista(nn) = {my(va = vector(nn)); va[1] = 1; my(vas = vecsort(va,,8)); for (n=2, nn, va[n] = nexta(va, vas, n-1); vas = vecsort(va,,8);); va;} \\ Michel Marcus, Feb 05 2019

cp's OEIS Frontend

A300907 a(n) is the least positive integer not yet in the sequence in which the largest digit of a(n-2) appears among its digits; a(1)=1, a(2)=2.

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A303294 a(n) is the least positive integer not yet in the sequence which shares a digit with either a(n-3) or a(n-2) (or with both), but shares no digit with a(n-1); a(1)=0, a(2)=1, a(3)=2.

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A302388 a(n) is the least positive integer not yet in the sequence in which the largest digit of a(n-3) appears among its digits; a(1)=1, a(2)=2, a(3)=3.

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PARI

A323708 a(n) is the smallest positive number not yet in the sequence that contains both the smallest and largest digits from a(n-1); a(1)=1.

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