A030283 -id:A030283 - cp's OEIS frontend

A054659 Increasing sequence with no repeating digits and no digits shared with previous term.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 23, 40, 51, 60, 71, 80, 91, 203, 415, 602, 713, 802, 913, 2045, 3167, 4025, 6137, 8024, 9135, 20467, 31589, 40267, 51389, 60247, 81359

Offset: 1

Henry Bottomley, Apr 18 2000

a(11)=23 since a(10)=10 and any number from 11 to 21 would share a digit between the two terms while 22 has a repeated digit

Cf. A030283.

def ok(s, t): return len(set(t)) == len(t) and len(set(s+t)) == len(s+t)
def agen(): # generator of complete sequence of terms
    an, MAX = 0, 987654321
    while True:
        if an < MAX: yield an
        else: return
        an, s = an+1, str(an)
        MAX = 10**(10-len(s))
        while an < MAX and not ok(s, str(an)): an += 1
print(list(agen())) # Michael S. Branicky, Jun 30 2022

A229363 a(1) = 0; for n > 1: a(n) = smallest even number greater than a(n-1) which does not use any digit used by a(n-1).

Original entry on oeis.org

0, 2, 4, 6, 8, 10, 22, 30, 42, 50, 62, 70, 82, 90, 112, 300, 412, 500, 612, 700, 812, 900, 1112, 3000, 4112, 5000, 6112, 7000, 8112, 9000, 11112, 30000, 41112, 50000, 61112, 70000, 81112, 90000, 111112, 300000, 411112, 500000, 611112, 700000, 811112, 900000

Offset: 1

Reinhard Zumkeller, Sep 21 2013

Essentially the same as A083490. - R. J. Mathar, Sep 24 2013

Index entries for linear recurrences with constant coefficients, signature (0, 1, 0, 0, 0, 0, 0, 10, 0, -10).

Cf. A030283, A229364, A005843.

import Data.List (intersect)
a229363 n = a229363_list !! (n-1)
a229363_list = f "" [0, 2 ..] where
   f xs (e:es) = if null $ intersect xs ys then e : f ys es else f xs es
                 where ys = show e

A358097 a(n) is the smallest integer m > n such that m and n have no common digit, or -1 when such integer m does not exist.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 22, 20, 30, 20, 20, 20, 20, 20, 20, 20, 31, 30, 30, 40, 30, 30, 30, 30, 30, 30, 41, 40, 40, 40, 50, 40, 40, 40, 40, 40, 51, 50, 50, 50, 50, 60, 50, 50, 50, 50, 61, 60, 60, 60, 60, 60, 70, 60, 60, 60, 71, 70, 70, 70, 70, 70, 70, 80, 70, 70, 81, 80, 80, 80, 80

Offset: 0

Bernard Schott, Oct 29 2022

When n is pandigital with or without 0 (A050278, A050289, A171102), m does not exist, so a(n) = -1; see examples for smallest pandigital cases.

			a(10) = 22; a(11) = 20; a(12) = 30.
a(123456789) = -1; a(1234567890) = -1.

Michel Marcus, Table of n, a(n) for n = 0..10000

Cf. A002275, A050278, A050289, A171102.
Cf. A030283 (trajectory starting 0).
Cf. A358098 (similar, with largest integer m < n).

a[n_] := Module[{d = Complement[Range[0, 9], IntegerDigits[n]], m = n + 1}, If[d == {} || d == {0}, -1, While[! AllTrue[IntegerDigits[m], MemberQ[d, #] &], m++]; m]]; Array[a, 100, 0] (* Amiram Eldar, Oct 29 2022 *)

isfull(d) = my(dd=setminus([0..9], d)); (dd==[]) || (dd==[0]);
a(n) = my(d=Set(digits(n))); if (isfull(d), -1, my(k=n+1); while (#setintersect(Set(digits(k)), d), k++); k); \\ Michel Marcus, Oct 29 2022

from itertools import count, product
def a(n):
    s = str(n)
    r = sorted(set("1234567890") - set(s))
    if len(r) == 0 or r == ["0"]: return -1
    for d in count(len(s)):
        for p in product(r, repeat=d):
            m = int("".join(p))
            if m > n: return m
print([a(n) for n in range(75)]) # Michael S. Branicky, Oct 29 2022

a(10^n-k) = 10^n when n >= 2 and 1 <= k <= 8.

a(10^n) = 2 * A002275(n+1), when n >= 1.

A030439 a(n+1) = smallest number not containing any digits of a(n), working in base 3.

Original entry on oeis.org

0, 1, 2, 3, 8, 9, 26, 27, 80, 81, 242, 243, 728, 729, 2186, 2187, 6560, 6561, 19682, 19683, 59048, 59049, 177146, 177147, 531440, 531441, 1594322, 1594323, 4782968, 4782969, 14348906, 14348907, 43046720, 43046721, 129140162, 129140163, 387420488, 387420489

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (0, 4, 0, -3).

Cf. A030283.

Bisections give: A024023, A000244.

3^i-1 then 3^i.
a(n) = 4*a(n-2)-3*a(n-4); g.f.: -x*(x^2-2*x-1) / ((x-1)*(x+1)*(3*x^2-1)). - Colin Barker, Sep 13 2014
a(n) = 3^floor(n/2) -1 +(n mod 2). - Alois P. Heinz, Sep 14 2014

A100373 Lexicographically earliest increasing sequence of composite numbers such that the digits of a(n) do not appear in a(n-1).

Original entry on oeis.org

4, 6, 8, 9, 10, 22, 30, 42, 50, 62, 70, 81, 90, 111, 200, 314, 500, 611, 700, 812, 900, 1111, 2000, 3111, 4000, 5111, 6000, 7111, 8000, 9111, 20000, 31111, 40000, 51111, 60000, 71111, 80000, 91111, 200000, 311113, 400000, 511112, 600000, 711111

Offset: 1

Labos Elemer, Dec 01 2004

Robert Israel, Table of n, a(n) for n = 1..7966

Cf. A002808, A030283, A030284, A030285, A030286, A030287, A030288, A030289, A030290.

f:= proc(x) local L,S,carry,m,nL,b,d0,Lz,z,i,d;
  L:= convert(x,base,10);
  nL:= nops(L);
  S:= sort(convert({$0..9} minus convert(L,set),list));
  b:= nops(S);
  d0:= min(select(`>`,S,L[-1]));
  if d0 = infinity then
    if S[1] = 0 then Lz:= Vector([0$nL, S[2]])
    else Lz:= Vector([S[1]$(nL+1)])
    fi
  else
    Lz:= Vector([S[1]$(nL-1),d0])
  fi;
  d:= LinearAlgebra:-Dimension(Lz);
  do
    z:= add(Lz[i]*10^(i-1),i=1..d);
    if not isprime(z) then return z fi;
    carry:= true;
    for i from 1 to d while carry do
      if Lz[i] = S[-1] then Lz[i]:= S[1]
      else
        carry:= false; if member(Lz[i],S,'m') then Lz[i]:= S[m+1] fi
      fi
    od;
    if carry then d:= d+1; if S[1] = 0 then Lz(d):= S[2] else Lz(d) := S[1] fi fi
  od;
end proc:
R:= 4: r:= 4:
for i from 2 to 100 do
  r:= f(r);
  R:= R,r
od:
R; # Robert Israel, Feb 27 2025

ta={1};Do[s1=IntegerDigits[Part[ta, Length[ta]]]; s2=IntegerDigits[n];If[Equal[Intersection[s1, s2], {}] &&!PrimeQ[n], Print[{Last[ta], n}];ta=Append[ta, n]], {n, 1, 1000000}];ta=Delete[ta, 1]

A049548 a(n+1) = smallest number not containing any digits of a(n), working in base 4.

Original entry on oeis.org

0, 1, 2, 3, 4, 10, 12, 21, 32, 53, 128, 213, 512, 853, 2048, 3413, 8192, 13653, 32768, 54613, 131072, 218453, 524288, 873813, 2097152, 3495253, 8388608, 13981013, 33554432, 55924053, 134217728, 223696213, 536870912, 894784853, 2147483648

Offset: 0

Henry Bottomley, Dec 28 2000

			Written in base 4 the sequence appears as 0, 1, 2, 3, 10, 22, 30, 111, 200, 311, 2000, 3111, 20000, 31111, 200000, 311111, 2000000, 3111111, etc. So a(9)=311 base 4 =53 base 10.

Index entries for linear recurrences with constant coefficients, signature (0,5,0,-4).

Cf. A030283, A030439.

LinearRecurrence[{0,5,0,-4},{0,1,2,3,4,10,12,21,32,53,128,213},40] (* Harvey P. Dale, Apr 27 2020 *)

For n>9, a(n)=4*a(n-2) + (a(n-2) mod 4).

a(n) = 5*a(n-2)-4*a(n-4) for n>5; g.f.: x*(32*x^10+16*x^9-12*x^8-12*x^7-17*x^6-x^4-6*x^3-2*x^2+2*x+1) / ((x-1)*(x+1)*(2*x-1)*(2*x+1)). - Colin Barker, Sep 13 2014

cp's OEIS Frontend

A054659 Increasing sequence with no repeating digits and no digits shared with previous term.

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A229363 a(1) = 0; for n > 1: a(n) = smallest even number greater than a(n-1) which does not use any digit used by a(n-1).

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A358097 a(n) is the smallest integer m > n such that m and n have no common digit, or -1 when such integer m does not exist.

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A030439 a(n+1) = smallest number not containing any digits of a(n), working in base 3.

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A100373 Lexicographically earliest increasing sequence of composite numbers such that the digits of a(n) do not appear in a(n-1).

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A049548 a(n+1) = smallest number not containing any digits of a(n), working in base 4.

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