A107353 -id:A107353 - cp's OEIS frontend

A107775 a(1)=4, a(n) = smallest integer not previously used which contains a digit from a(n-1).

4, 14, 1, 10, 0, 20, 2, 12, 11, 13, 3, 23, 21, 15, 5, 25, 22, 24, 26, 6, 16, 17, 7, 27, 28, 8, 18, 19, 9, 29, 32, 30, 31, 33, 34, 35, 36, 37, 38, 39, 43, 40, 41, 42, 44, 45, 46, 47, 48, 49, 54, 50, 51, 52, 53, 55, 56, 57, 58, 59, 65, 60, 61, 62, 63, 64, 66, 67, 68, 69, 76, 70

Offset: 1

Eric Angelini & Zak Seidov, May 24 2005

Cf. A107353 a(1)=0, A107772 a(1)=1, A107773 a(1)=2, A107774 a(1)=3, A107776 a(1)=5, A107777 a(1)=6, A107778 a(1)=7, A107779 a(1)=8, A107780 a(1)=9, A107781 a(1)=10

Cf. A107353, A107772, A107773, A107774, A107776, A107777, A107778, A107779, A107780, A107781.

f[l_] := Block[{c = 0}, While[ MemberQ[l, c] || Intersection @@ IntegerDigits /@{Last[l], c}=={}, c++ ];Return[Append[l, c]]];Nest[f, {4}, 70] (* Ray Chandler, Jul 19 2005 *)

A107776 a(1)=5, a(n) = smallest integer not previously used which contains a digit from a(n-1).

Original entry on oeis.org

5, 15, 1, 10, 0, 20, 2, 12, 11, 13, 3, 23, 21, 14, 4, 24, 22, 25, 26, 6, 16, 17, 7, 27, 28, 8, 18, 19, 9, 29, 32, 30, 31, 33, 34, 35, 36, 37, 38, 39, 43, 40, 41, 42, 44, 45, 46, 47, 48, 49, 54, 50, 51, 52, 53, 55, 56, 57, 58, 59, 65, 60, 61, 62, 63, 64, 66, 67, 68, 69, 76, 70

Offset: 1

Eric Angelini & Zak Seidov, May 24 2005

Cf. A107353 a(1)=0, A107772 a(1)=1, A107773 a(1)=2, A107774 a(1)=3, A107775 a(1)=4, A107777 a(1)=6, A107778 a(1)=7, A107779 a(1)=8, A107780 a(1)=9, A107781 a(1)=10

Cf. A107353, A107772, A107773, A107774, A107775, A107777, A107778, A107779, A107780, A107781.

f[l_] := Block[{c = 0}, While[ MemberQ[l, c] || Intersection @@ IntegerDigits /@{Last[l], c}=={}, c++ ];Return[Append[l, c]]];Nest[f, {5}, 70] (* Ray Chandler, Jul 19 2005 *)

A107777 a(1)=6, a(n) = smallest integer not previously used which contains a digit from a(n-1).

Original entry on oeis.org

6, 16, 1, 10, 0, 20, 2, 12, 11, 13, 3, 23, 21, 14, 4, 24, 22, 25, 5, 15, 17, 7, 27, 26, 28, 8, 18, 19, 9, 29, 32, 30, 31, 33, 34, 35, 36, 37, 38, 39, 43, 40, 41, 42, 44, 45, 46, 47, 48, 49, 54, 50, 51, 52, 53, 55, 56, 57, 58, 59, 65, 60, 61, 62, 63, 64, 66, 67, 68, 69, 76, 70

Offset: 1

Eric Angelini & Zak Seidov, May 24 2005

Cf. A107353 a(1)=0, A107772 a(1)=1, A107773 a(1)=2, A107774 a(1)=3, A107775 a(1)=4, A107776 a(1)=5, A107778 a(1)=7, A107779 a(1)=8, A107780 a(1)=9, A107781 a(1)=10

Cf. A107353, A107772, A107773, A107774, A107775, A107776, A107778, A107779, A107780, A107781.

f[l_] := Block[{c = 0}, While[ MemberQ[l, c] || Intersection @@ IntegerDigits /@{Last[l], c}=={}, c++ ];Return[Append[l, c]]];Nest[f, {6}, 70] (* Ray Chandler, Jul 19 2005 *)

A107779 a(1)=8, a(n) = smallest integer not previously used which contains a digit from a(n-1).

Original entry on oeis.org

8, 18, 1, 10, 0, 20, 2, 12, 11, 13, 3, 23, 21, 14, 4, 24, 22, 25, 5, 15, 16, 6, 26, 27, 7, 17, 19, 9, 29, 28, 32, 30, 31, 33, 34, 35, 36, 37, 38, 39, 43, 40, 41, 42, 44, 45, 46, 47, 48, 49, 54, 50, 51, 52, 53, 55, 56, 57, 58, 59, 65, 60, 61, 62, 63, 64, 66, 67, 68, 69, 76, 70

Offset: 1

Eric Angelini & Zak Seidov, May 24 2005

Cf. A107353 a(1)=0, A107772 a(1)=1, A107773 a(1)=2, A107774 a(1)=3, A107775 a(1)=4, A107776 a(1)=5, A107777 a(1)=6, A107778 a(1)=7, A107780 a(1)=9, A107781 a(1)=10

Cf. A107353, A107772, A107773, A107774, A107775, A107776, A107777, A107778, A107780, A107781.

f[l_] := Block[{c = 0}, While[ MemberQ[l, c] || Intersection @@ IntegerDigits /@{Last[l], c}=={}, c++ ];Return[Append[l, c]]];Nest[f, {8}, 70] (* Ray Chandler, Jul 19 2005 *)

A107781 a(1)=10, a(n) = smallest integer not previously used which contains a digit from a(n-1).

Original entry on oeis.org

10, 0, 20, 2, 12, 1, 11, 13, 3, 23, 21, 14, 4, 24, 22, 25, 5, 15, 16, 6, 26, 27, 7, 17, 18, 8, 28, 29, 9, 19, 31, 30, 32, 33, 34, 35, 36, 37, 38, 39, 43, 40, 41, 42, 44, 45, 46, 47, 48, 49, 54, 50, 51, 52, 53, 55, 56, 57, 58, 59, 65, 60, 61, 62, 63, 64, 66, 67, 68, 69, 76, 70

Offset: 1

Eric Angelini & Zak Seidov, May 24 2005

Cf. A107353 a(1)=0, A107772 a(1)=1, A107773 a(1)=2, A107774 a(1)=3, A107775 a(1)=4, A107776 a(1)=5, A107777 a(1)=6, A107778 a(1)=7, A107779 a(1)=8, A107780 a(1)=9

Cf. A107353, A107772, A107773, A107774, A107775, A107776, A107777, A107778, A107779, A107780.

f[l_] := Block[{c = 0}, While[ MemberQ[l, c] || Intersection @@ IntegerDigits /@{Last[l], c}=={}, c++ ];Return[Append[l, c]]];Nest[f, {10}, 70] (* Ray Chandler, Jul 19 2005 *)

A362371 a(0)=0. For each digit in the sequence, append the smallest unused integer that contains that digit.

Original entry on oeis.org

0, 10, 1, 20, 11, 2, 30, 12, 13, 21, 3, 40, 14, 22, 15, 23, 24, 16, 31, 4, 50, 17, 34, 25, 26, 18, 5, 27, 32, 28, 41, 19, 6, 33, 51, 42, 35, 60, 61, 7, 36, 43, 29, 45, 52, 46, 71, 8, 53, 62, 37, 38, 72, 82, 48, 44, 81, 91, 9, 56, 39, 63, 54, 100, 47, 92, 73

Offset: 0

Gavin Lupo, Apr 17 2023

			a(0) =  0
a(1) = 10 (from digit 0 in a(0)=0, smallest integer other than 0).
a(2) =  1 (from digit 1 in a(1)=10, smallest integer other than 10).
a(3) = 20 (from digit 0 in a(1)=10, smallest integer other than 0 and 10).
a(4) = 11 (from digit 1 in a(2)=1, smallest integer other than 1 and 10).
a(5) =  2 (from digit 2 in a(3)=20, smallest integer other than 20).
a(6) = 30 (from digit 0 in a(3)=20, smallest integer other than 0, 10, and 20).

Gavin Lupo, Image of the first 10000 terms

Cf. A000002, A107353, A339853.

from itertools import count, islice
def agen(): # generator of terms
    s, aset, mink = "0", set(), 0
    for n in count(0):
        an = mink
        while an in aset or set(san:=str(an)) & {s[0]} == set(): an += 1
        s = s[1:] + san
        aset.add(an)
        yield an
        while mink in aset: aset.discard(mink); mink += 1
print(list(islice(agen(), 67))) # Michael S. Branicky, Apr 25 2023

A317330 a(n) is the smallest positive integer not yet in the sequence that contains a digit equal to the sum of the digits of a(n-1) (mod 10); a(1)=0.

Original entry on oeis.org

0, 10, 1, 11, 2, 12, 3, 13, 4, 14, 5, 15, 6, 16, 7, 17, 8, 18, 9, 19, 20, 21, 23, 25, 27, 29, 31, 24, 26, 28, 30, 32, 35, 38, 41, 45, 39, 22, 34, 37, 40, 42, 36, 49, 33, 46, 50, 51, 56, 61, 47, 71, 48, 52, 57, 62, 58, 43, 67, 53, 68, 44, 78, 54, 59, 64, 60, 63, 69, 55, 70, 72, 79

Offset: 1

Enrique Navarrete, Jul 25 2018

Up to n=150 the only consecutive terms in the sequence are 19,20,21; 50,51; 90,91; 100,101; 106,107; 108,109,110.
Up to n=150 the sequence of first differences is bounded by -57 and 57 (in nonconsecutive terms).
From Robert G. Wilson v, Jul 26 2018: (Start)
It appears that every number appears.
If so the inverse permutation would be: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 20, 21, 37, 22, 27, 23, ..., .
(End)
Yes, every number appears.  Every pandigital number must eventually appear, and for each d in [0,9] there are infinitely many pandigital numbers with digit sum == d (mod 10), so every number containing digit d will eventually appear. - Robert Israel, Aug 30 2018

			a(5)=2 since a(4)=11 and 1+1 is congruent to 2 (mod 10).
a(21)=20 since a(20)=19 and 1+9 is congruent to 0 (mod 10).

Robert G. Wilson v, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Cf. A107353.

N:= 1000: # to get all terms before the first term > N
A[1]:= 0:
for d from 0 to 9 do S[d]:= select(t -> member(d, convert(t,base,10)), {$1..N}) od:
for n from 2 do
  dd:= convert(convert(A[n-1],base,10),`+`) mod 10;
  if S[dd] = {} then break fi;
  A[n]:= min(S[dd]);
  for d from 0 to 9 do S[d]:= S[d] minus {A[n]} od:
od:
seq(A[i],i=1..n-1); # Robert Israel, Aug 30 2018

f[lst_List] := Block[{k = 1, l = Mod[Plus @@ IntegerDigits@lst[[-1]], 10]}, While[MemberQ[lst, k] || Union[MemberQ[{l}, #] & /@ IntegerDigits@k][[-1]] == False, k++]; Append[lst, k]]; Nest[f, {0}, 72] (* Robert G. Wilson v, Jul 26 2018 *)

A298208 a(n) is the smallest positive integer not yet in the sequence that shares a digit with a(n-2) and shares no digit with a(n-1); a(1) = 0, a(2) = 1.

Original entry on oeis.org

0, 1, 20, 11, 2, 10, 22, 13, 24, 3, 4, 23, 14, 25, 16, 5, 6, 15, 26, 17, 28, 7, 8, 27, 18, 29, 31, 9, 12, 39, 21, 30, 19, 32, 41, 33, 40, 35, 42, 36, 44, 37, 45, 38, 46, 53, 47, 50, 34, 51, 43, 52, 48, 55, 49, 56, 74, 58, 64, 57, 60, 54, 61, 59, 62, 75, 63, 70

Offset: 1

Enrique Navarrete, Jan 15 2018

Initial fixed points are 47, 52, 56, 58, 72, 81, 94, 101, 13661, 13663. - Corrected and extended by Robert Israel, Feb 09 2018

Inverse: 0, 1, 4, 9, 10, 15, 16, 21, 22, 27, 5, 3, 28, 7, 12, 17, 14, 19, 24, 32, 2, 30, 6, 11, 8, 13, 18, 23, 20, 25, 31, ..., . - Robert G. Wilson v, Feb 09 2018

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A107353 (where each term must share a digit with the preceding term).

Cf. A297418.

N:= 1000: # to get all terms before the first term > N
a[1] := 0: a[2] := 1: first := 2:
Next := Array(2 .. N, i -> i+1):
Prev := Array(2 .. N, i -> i-1): Prev[2] := 0:
for n from 0 to N do
  digs[n] := convert(convert(n, base, 10), set)
od:
for n from 3 do
  D1 := digs[a[n-1]];
  D2 := digs[a[n-2]];
  t := first;
  while digs[t] intersect D2 = {} or digs[t] intersect D1 <> {} do
    t := Next[t];
    if t > N then break fi
  od;
  if t > N then break fi;
  a[n] := t;
  if Prev[t] = 0 then first := Next[t] else Next[Prev[t]] := Next[t] fi; if Next[t] <= N then Prev[Next[t]] := Prev[t] fi
od:
seq(a[i],i=1..n-1); # Robert Israel, Feb 09 2018

f[s_List] := Block[{a = Union@ IntegerDigits@ s[[-2]], b = Union@ IntegerDigits@ s[[-1]], k = 2}, While[id = Union@ IntegerDigits@ k; MemberQ[s, k] || Intersection[a, id] == {} || Intersection[b, id] != {}, k++]; Append[s, k]]; Nest[f, {0, 1}, 66] (* Robert G. Wilson v, Feb 09 2018 *)

A302095 a(n) is the smallest positive integer not yet in the increasing sequence that is obtained when the largest digit from a(n-1) is deleted and the remaining digits are permuted such that no digit in a(n) has the same position it had in a(n-1) (counting from left to right). No repeated digits allowed; a(1)=10.

Original entry on oeis.org

10, 230, 402, 520, 602, 720, 802, 920, 1023, 2104, 3012, 4120, 5012, 6120, 7012, 8120, 9012, 12034, 20153, 31024, 50132, 61023, 70132, 81023, 90132, 120435, 201346, 310254, 401326, 510234, 601342, 710234, 801342, 910234, 1023456, 2104375, 3012456, 4103275, 5012346, 7103254

Offset: 1

Enrique Navarrete, May 19 2018

All terms in the sequence contain 0.
The fact that all digits in the terms are distinct makes the sequence finite.
In fact, the sequence contains 59 terms and a(59)=901325476.
The terms that require the smallest number of permutations to recover their natural ordering are a(1)=10, a(9)=1023 and a(35)=1023456 (one permutation required).

			a(2)=230 since it is the smallest positive integer not yet in the sequence that is obtained when the largest digit 1 from a(1)=10 is deleted, the remaining digit 0 is permuted from the second to third place, and no digits are repeated.

Enrique Navarrete, Table of n, a(n) for n = 1..59

Cf. A107353, A297418.

A303657 a(n) is the least positive integer not yet in the sequence which shares a digit with a(n-2); a(1)=0, a(2)=1.

Original entry on oeis.org

0, 1, 10, 11, 12, 13, 2, 3, 20, 23, 21, 22, 14, 24, 4, 25, 34, 5, 30, 15, 31, 16, 17, 6, 7, 26, 27, 28, 29, 8, 9, 18, 19, 38, 39, 32, 33, 35, 36, 37, 43, 47, 40, 41, 42, 44, 45, 46, 48, 49, 54, 59, 50, 51, 52, 53, 55, 56, 57, 58, 65, 68, 60, 61, 62, 63, 64, 66

Offset: 1

Enrique Navarrete, Apr 27 2018

Up to n=34 the first differences of the sequence are bounded by -29 and 25; the bounds are -10 and 10 thereafter.
From a(103)=100 onwards, the sequence has slope = 1 with minor jumps about every 100 terms.
It appears that this sequence has an inverse, namely 1, 2, 7, 8, 15, 18, 24, 25, 30, 31, 3, 4, 5, 6, 13, 20, 22, 23, 32, 33, 9, ..., . - Robert G. Wilson v, Apr 29 2018 [Edited by Rémy Sigrist, May 06 2018]

			a(7)=2 since it is the least positive integer not yet in the sequence which shares a digit with a(5)=12.

R. J. Cano, Sequencer program in PARI.

Cf. A107353, A303605.

f[s_List] := Block[{k = 2, l = Union@ IntegerDigits@ s[[-2]]}, While[MemberQ[s, k] || Intersection[l, IntegerDigits@ k] == {}, k++]; Append[s, k]]; Nest[f, {0, 1}, 70] (* Robert G. Wilson v, Apr 29 2018 *)

Digits(x,b)=if(!x,[0],digits(x,b));
firstTerms(n,{k=2},{b=10})={my(N=b*n);my(s=List(vector(N,u,u-1)),t,x,y);for(m=k+1,n,x=Set(Digits(s[m-k],b));for(i=m,N,y=Set(Digits(s[i],b));if(#setintersect(x,y),t=s[i];listpop(s,i);listinsert(s,t,m);break)));return(Vec(s)[1..n])}
a(n)=firstTerms(n)[n]; \\ R. J. Cano, May 05 2018

PARI
```
See Cano link.
```

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A107775 a(1)=4, a(n) = smallest integer not previously used which contains a digit from a(n-1).

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A107776 a(1)=5, a(n) = smallest integer not previously used which contains a digit from a(n-1).

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A107777 a(1)=6, a(n) = smallest integer not previously used which contains a digit from a(n-1).

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A107779 a(1)=8, a(n) = smallest integer not previously used which contains a digit from a(n-1).

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A107781 a(1)=10, a(n) = smallest integer not previously used which contains a digit from a(n-1).

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A362371 a(0)=0. For each digit in the sequence, append the smallest unused integer that contains that digit.

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A317330 a(n) is the smallest positive integer not yet in the sequence that contains a digit equal to the sum of the digits of a(n-1) (mod 10); a(1)=0.

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A298208 a(n) is the smallest positive integer not yet in the sequence that shares a digit with a(n-2) and shares no digit with a(n-1); a(1) = 0, a(2) = 1.

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A303657 a(n) is the least positive integer not yet in the sequence which shares a digit with a(n-2); a(1)=0, a(2)=1.

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