A300907 -id:A300907 - cp's OEIS frontend

A302566 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, and a(n) cannot share digits with a(n-1); a(1) = 1, a(2) = 2.

1, 2, 10, 22, 11, 20, 13, 24, 3, 4, 23, 14, 30, 41, 32, 40, 31, 42, 33, 44, 35, 46, 5, 6, 15, 26, 45, 16, 25, 36, 50, 61, 52, 60, 51, 62, 53, 64, 55, 63, 54, 66, 57, 68, 7, 8, 17, 28, 37, 18, 27, 38, 47, 58, 67, 48, 70, 81, 72, 80, 71, 82, 73, 84, 75, 83, 74, 85, 76, 88, 77, 86, 79, 108, 9, 78, 19, 87

Offset: 1

Enrique Navarrete, Apr 09 2018

The only pairs of consecutive numbers are 1, 2; 3, 4; 5, 6; 7, 8.
The 2-digit numbers that are not in the sequence are 12, 21, 34, 43, 56, 65, 89, 98 (78 and 87 do appear at n = 76 and n = 78, respectively).
The last 2-digit number to appear is 91 at n = 113.
Starting from a(73) = 79, the first term of alternating terms must contain 9, and the second term must contain 8.
If the initial terms of this sequence are swapped (by defining a(1)=2 and a(2)=1) then the resulting sequence is identical to this for n>12. - R. J. Cano, Apr 13 2018

			a(4) = 22 since it contains the largest digit from a(2) = 2, and can't be 12 since the digit 1 appears in a(3) = 10.

Rémy Sigrist, PARI program for A302566
R. J. Cano, Sequencer program in PARI

Cf. A011531-A011540, A300907.

Nest[Append[#, Block[{k = 2, d}, While[Nand[FreeQ[#[[All, 1]], k], MemberQ[Set[d, IntegerDigits[k]], Max[#[[-2, -1]] ] ], ! IntersectingQ[d, #[[-1, -1]]] ], k++]; {k, d}]] &, Transpose@ {#, IntegerDigits@ #} &@ Range[2], 73][[All, 1]] (* Michael De Vlieger, Apr 12 2018 *)

PARI
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\\ See Links section.
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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.

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

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A302566 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, and a(n) cannot share digits with a(n-1); 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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