A276633 -id:A276633 - cp's OEIS frontend

A067581 a(n) = smallest integer not yet in the sequence with no digits in common with a(n-1), a(0)=0.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 22, 11, 20, 13, 24, 15, 23, 14, 25, 16, 27, 18, 26, 17, 28, 19, 30, 12, 33, 21, 34, 29, 31, 40, 32, 41, 35, 42, 36, 44, 37, 45, 38, 46, 39, 47, 50, 43, 51, 48, 52, 49, 53, 60, 54, 61, 55, 62, 57, 63, 58, 64, 59, 66, 70, 56, 71, 65, 72, 68, 73, 69

Offset: 0

Ulrich Schimke (ulrschimke(AT)aol.com)

David W. Wilson has shown that the sequence contains every positive integer except those containing all the digits 1 through 9 (which obviously have no possible predecessor). Jun 04 2002

a(A137857(n)) = A137857(n). - Reinhard Zumkeller, Feb 15 2008

			a(14) = 13, since a(13) = 20 and all integers smaller than 13 have a digit in common with 20 or have already appeared in the sequence.

Reinhard Zumkeller and Zak Seidov, Table of n, a(n) for n = 0..10000

Cf. A136332, A184992, A239664, A249585, A249591.

A342441 a(1) = 1; for n > 1, a(n) is the least positive integer not occurring earlier such that a(n-1)+a(n) shares no digit with either a(n-1) or a(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 14, 15, 17, 16, 18, 12, 21, 19, 25, 22, 23, 26, 24, 27, 28, 29, 31, 33, 32, 34, 35, 36, 38, 39, 41, 42, 43, 37, 44, 45, 46, 47, 48, 51, 149, 53, 49, 52, 54, 55, 56, 57, 59, 58, 63, 64, 65, 66, 67, 68, 62, 69, 72, 73, 75, 85, 77, 74, 76, 78, 82, 79

Offset: 1

Scott R. Shannon, Mar 12 2021

No term can end in 0 as that would result in the last digit of a(n-1) being the same as the last digit of a(n-1)+a(n).

			a(2) = 2 as a(1)+2 = 1+2 = 3 which shares no digit with a(1) = 1 or 2.
a(10) = 11 as a(9)+11 = 9+11 = 20 which shares no digit with a(9) = 9 or 11. Note that the first number skipped is 10 as 9+10 = 19 which shares a digit with 9.
a(11) = 13 as a(10)+13 = 11+13 = 24 which shares no digit with a(10) = 11 or 13. Note that the number 12 is skipped as 11+12 = 23 which shares a digit with 12.

Scott R. Shannon, Image of the first 100000 terms. The green line is a(n) = n.

Cf. A342442 (multiplication), A276633, A010784, A043537, A043096, A338466, A336285.

Block[{a = {1}, m = {1}, d, s, k}, Do[k = 2; While[Nand[FreeQ[a, k], ! IntersectingQ[Set[d, IntegerDigits[k]], Set[s, IntegerDigits[a[[-1]] + k]]], ! IntersectingQ[s, m]], k++]; AppendTo[a, k]; Set[m, d], 72]; a] (* Michael De Vlieger, Mar 20 2021 *)

def aupton(terms):
  alst, aset = [1], {1}
  while len(alst) < terms:
    an, anm1_digs = 2, set(str(alst[-1]))
    while True:
      while an in aset: an += 1
      if (set(str(an)) | anm1_digs) & set(str(an+alst[-1])) == set():
        alst.append(an); aset.add(an); break
      an += 1
  return alst
print(aupton(73)) # Michael S. Branicky, Mar 20 2021

A336285 a(0) = 0; for n > 0, a(n) is the least positive integer not occurring earlier such that the digits in a(n-1)+a(n) are all distinct.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 53, 50, 52, 51, 54, 55, 65, 58, 62, 61, 59, 64, 56, 67, 57, 63, 60, 66, 68, 69, 70, 72, 71, 74, 73, 75, 77

Offset: 0

Rémy Sigrist, Jul 22 2020

In other words, for any n > 0, a(n) + a(n+1) belongs to A010784.

The sequence is finite since there are only a finite number of positive integers with distinct digits, see A010784, although the exact number of terms is currently unknown.

			The first terms, alongside a(n) + a(n+1), are:
  n   a(n)  a(n)+a(n+1)
  --  ----  -----------
   0     0            1
   1     1            3
   2     2            5
   3     3            7
   4     4            9
   5     5           12
   6     7           13
   7     6           14
   8     8           17
   9     9           19
  10    10           21

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Scott R. Shannon, Image of the first 1000000 terms. The green line is a(n) = n.

Cf. A342383, A338466, A322845, A010784, A043537, A043096, A276633, A002378.

s=0; v=1; for (n=1, 67, print1 (v", "); s+=2^v; for (w=1, oo, if (!bittest(s, w) && #(d=digits(v+w))==#Set(d), v=w; break)))

def agen():
  alst, aset, min_unused = [0], {0}, 1
  yield 0
  while True:
    an = min_unused
    while True:
      while an in aset: an += 1
      t = str(alst[-1] + an)
      if len(t) == len(set(t)):
        alst.append(an); aset.add(an); yield an
        if an == min_unused: min_unused = min(set(range(max(aset)+2))-aset)
        break
      an += 1
g = agen()
print([next(g) for n in range(77)]) # Michael S. Branicky, Mar 11 2021

a(0)=0 added by N. J. A. Sloane, Mar 14 2021

A338466 a(0) = 0; for n > 0, a(n) is the least positive integer not occurring earlier such that the digits in a(n-1)*a(n) are all distinct.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 11, 13, 14, 15, 16, 19, 18, 17, 20, 21, 22, 23, 26, 24, 27, 25, 29, 28, 30, 31, 33, 32, 39, 34, 37, 35, 36, 38, 40, 41, 43, 42, 45, 44, 47, 50, 49, 52, 48, 55, 46, 51, 53, 56, 54, 57, 60, 58, 62, 59, 66, 61, 64, 63, 65, 71, 70, 67, 69, 68, 72, 74, 73, 77, 79

Offset: 0

Scott R. Shannon, Mar 09 2021

The sequence is finite, the 71782nd term being a(71781) = 50005 beyond which no number exists that has not occurred earlier such that 50005*a(n) has distinct digits. The maximum term is a(71428) = 175446.

			a(1) = 1 as a(0)*1 = 0*1 = 0 which has one distinct digit 0.
a(10) = 10 as a(9)*10 = 9*10 = 90 which has two distinct digits 9 and 0.
a(11) = 12 as a(10)*12 = 10*12 = 120 which has three distinct digits. Note that 11 is the first skipped number as 10*11 = 110 which has 1 as a duplicate digit.
a(12) = 11 as a(11)*11 = 12*11 = 132 which has three distinct digits.

Scott R. Shannon, Image of the 71782 terms. The green line is a(n) = n.

Cf. A342382, A010784, A043537, A043096, A276633, A002378.

Offset corrected by N. J. A. Sloane, Jun 16 2021

A342442 a(1) = 2; for n > 1, a(n) is the least positive integer not occurring earlier such that a(n-1)*a(n) shares no digit with either a(n-1) or a(n).

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 42, 14, 17, 18, 15, 16, 13, 19, 32, 22, 23, 26, 29, 12, 25, 24, 34, 27, 33, 36, 39, 43, 37, 38, 28, 47, 44, 45, 46, 63, 66, 65, 48, 49, 62, 55, 54, 35, 174, 53, 76, 57, 56, 59, 52, 58, 64, 92, 74, 68, 78, 72, 77, 67, 73, 83, 69, 79, 84, 75, 88, 113, 183, 138, 149, 148

Offset: 1

Scott R. Shannon, Mar 12 2021

No term can end in 0 or 1 as that would result in the last digit of a(n-1)*a(n) being the same as a(n)'s last digit. The majority of terms appear to grow linearly with n but occasional large spikes in the values also occur, e.g. a(47888) = 425956849. See the examples. It is unknown if the sequence is infinite.

			a(2) = 3 as a(1)*3 = 2*3 = 6 which shares no digit with a(1) = 2 or 3.
a(9) = 42 as a(8)*42 = 9*42 = 378 which shares no digit with a(8) = 9 or 42.
a(10) = 14 as a(9)*14 = 42*14 = 588 which shares no digit with a(9) = 42 or 14.
a(47888) = 425956849 as a(47887)*425956849 = 258649*425956849 = 110173313037001 which shares no digit with a(47887) = 258649 or 425956849.

Daniel Mondot, Table of n, a(n) for n = 1..10000

Cf. A342441 (addition), A342755, A276633, A010784, A043537, A043096, A338466, A336285.

def aupton(terms):
  alst, aset = [2], {2}
  while len(alst) < terms:
    an, anm1_digs = 2, set(str(alst[-1]))
    while True:
      while an in aset: an += 1
      if (set(str(an)) | anm1_digs) & set(str(an*alst[-1])) == set():
        alst.append(an); aset.add(an); break
      an += 1
  return alst
print(aupton(74)) # Michael S. Branicky, Mar 20 2021

A239664 a(n) = 1, a(n+1) = smallest number not occurring earlier, having with a(n) neither a common digit nor a common divisor.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 23, 11, 20, 13, 22, 15, 26, 17, 24, 19, 25, 14, 27, 16, 29, 18, 35, 12, 37, 21, 34, 55, 28, 31, 40, 33, 41, 30, 47, 32, 45, 38, 49, 36, 59, 42, 53, 44, 39, 46, 51, 43, 50, 61, 48, 65, 71, 52, 63, 58, 67, 54, 73, 56, 79, 60, 77

Offset: 1

Reinhard Zumkeller, Mar 23 2014

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A054659, A067581, A276633, A276512.

import Data.List (delete, intersect); import Data.Function (on)
a239664 n = a239664_list !! (n-1)
a239664_list = 1 : f 1 [2..] where
   f v ws = g ws where
     g (x:xs) = if gcd v x == 1 && ((intersect `on` show) v x == "")
                   then x : f x (delete x ws) else g xs

{u=[]; a=1; for(n=1,99, print1(a","); u=setunion(u,[a]); while(#u>1&&u[2]==u[1]+1,u=u[^1]); for(k=u[1]+1,9e9, setsearch(u,k)&&next;gcd(k,a)>1&&next; #setintersect(Set(digits(a)),Set(digits(k)))&&next; a=k; next(2)));a} \\ M. F. Hasler, Sep 17 2016

A276512 a(n) = smallest integer not yet in the sequence with no digits in common with a(n-2); a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 22, 20, 13, 14, 24, 23, 15, 16, 26, 25, 17, 18, 28, 27, 19, 30, 32, 12, 40, 33, 21, 29, 34, 31, 50, 42, 36, 35, 41, 44, 37, 38, 45, 46, 39, 51, 47, 43, 52, 55, 48, 49, 53, 56, 60, 70, 54, 58, 61, 62, 57, 59, 63, 64, 71, 72, 65, 66, 73, 74, 68, 69, 75

Offset: 0

Zak Seidov and Eric Angelini, Sep 06 2016

This is not a permutation of the nonnegative integers. E.g. 123456789 and 1023456789 (the smallest pandigital number) are not members.

a(n) = n for n = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 34, 84, 104, 105, 1449, 2889, 3183, ...

Zak Seidov, Table of n, a(n) for n = 0..5000

Cf. A054659, A067581, A276633, A276766.

s={0,1};Do[a=s[[-2]];n=2; While[MemberQ[s,n]||Intersection [IntegerDigits[a],IntegerDigits[n]]≠{}, n++];AppendTo[s,n],{100}];s

from itertools import count, islice, product as P
def only(s, D=1): # numbers with >= D digits only from s
    yield from (int("".join(p)) for d in count(D) for p in P(s, repeat=d))
def agen(): # generator of terms
    aset, an1, an, minan = {0, 1}, 0, 1, 2
    yield from [0, 1]
    while True:
        an1, an, s = an, minan, set(str(an1))
        use = "".join(c for c in "0123456789" if c not in s)
        for an in only(use, D=len(str(minan))):
            if an not in aset: break
        aset.add(an)
        yield an
        while minan in aset: minan += 1
print(list(islice(agen(), 75))) # Michael S. Branicky, Jun 30 2022

A342382 a(0) = 0; for n > 0, a(n) is the least positive integer not occurring earlier such that both the digits in a(n) and the digits in a(n-1)*a(n) are all distinct.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 19, 18, 17, 20, 21, 23, 26, 24, 27, 25, 29, 28, 30, 31, 34, 37, 35, 36, 38, 39, 32, 40, 41, 43, 42, 45, 48, 52, 49, 50, 47, 51, 46, 53, 56, 54, 57, 60, 58, 62, 59, 68, 64, 61, 65, 63, 72, 69, 67, 70, 71, 73, 74, 76, 78, 80, 79, 82, 75, 81, 83

Offset: 0

Scott R. Shannon, Mar 09 2021

The sequence is finite, the 18351st term being a(18350) = 41987 beyond which no number exists that has not occurred earlier that has all distinct digits and that 41987*a(n) has all distinct digits. The maximum term is a(18097) = 219087.

			a(1) = 1 as 1 has one distinct digit and a(0)*1 = 0*1 = 0 which has one distinct digit 0.
a(10) = 10 as 10 has two distinct digits and a(9)*10 = 9*10 = 90 which has two distinct digits 9 and 0.
a(11) = 12 as 12 has two distinct digits and a(10)*12 = 10*12 = 120 which has three distinct digits. Note that 11 is the first skipped number as 11 has 1 as a duplicate digit.
a(16) = 19 as 19 has two distinct digits and a(15)*19 = 16*19 = 304 which has three distinct digits. Note that 17 and 18 are skipped as 16*17 = 272 while 16*18 = 288, both of which contain duplicate digits.

Scott R. Shannon, Image of the 18351 terms. The green line is a(n) = n.

Cf. A338466, A010784, A043537, A043096, A276633, A002378.

Block[{a = {0}, k, m = 42000}, Do[k = 1; While[Nand[FreeQ[a, k], AllTrue[DigitCount[a[[-1]]*k], # < 2 &], AllTrue[DigitCount[k], # < 2 &]], If[k > m, Break[]]; k++]; If[k > m, Break[]]; AppendTo[a, k], {i, 76}]; a] (* Michael De Vlieger, Mar 11 2021 *)

Offset corrected by N. J. A. Sloane, Jun 16 2021

A276766 a(n) = smallest nonnegative integer not yet in the sequence with no repeated digits and no digits in common with a(n-1), starting with a(0)=0.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 23, 14, 20, 13, 24, 15, 26, 17, 25, 16, 27, 18, 29, 30, 12, 34, 19, 28, 31, 40, 21, 35, 41, 32, 45, 36, 42, 37, 46, 38, 47, 39, 48, 50, 43, 51, 49, 52, 60, 53, 61, 54, 62, 57, 63, 58, 64, 59, 67, 80, 56, 70, 65, 71, 68, 72, 69, 73, 81, 74, 82, 75

Offset: 0

Claudio Meller, Sep 17 2016

The author of this sequence is Rodolfo Kurchan, who mentioned this sequence in a Facebook Group "Series", cf. link.

The sequence is finite, with last term a(5274) = 78642. - M. F. Hasler, Sep 17 2016

M. F. Hasler, Table of n, a(n) for n = 0..5274 (All terms)
Rodolfo Kurchan, Post in Facebook Group "Series".

Cf. A054659, A067581, A276633, A276512.

{u=[]; (t(k)=if(#Set(k=digits(k))==#k,k)); a=1; for(n=1, 99, print1(a","); u=setunion(u, [a]); t(u[1])||u[1]++; while(#u>1&&u[2]<=u[1]+1, u=u[^1]); for(k=u[1]+1, 9e9, setsearch(u, k)&&next; (d=t(k))&& !#setintersect(Set(digits(a)), Set(d))&&(a=k)&&next(2))); a} \\ M. F. Hasler, Sep 17 2016

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
    aset, k, mink, MAX = {0}, 0, 1, 987654321
    while True:
        if k < MAX: yield k
        else: return
        k, s = mink, str(k)
        MAX = 10**(10-len(s))
        while k < MAX and (k in aset or not ok(s, str(k))):
            k += 1
        aset.add(k)
        while mink in aset: mink += 1
print(list(agen())[:73]) # Michael S. Branicky, Jun 30 2022

Edited by M. F. Hasler, Sep 17 2016

A342383 a(0) = 0; for n > 0, a(n) is the least positive integer not occurring earlier such that both the digits in a(n) and the digits in a(n-1)+a(n) are all distinct.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 6, 8, 9, 10, 13, 12, 14, 15, 16, 18, 17, 19, 20, 21, 24, 23, 25, 26, 27, 29, 28, 30, 31, 32, 35, 34, 36, 37, 38, 40, 39, 41, 42, 43, 46, 45, 47, 48, 49, 53, 50, 52, 51, 54, 69, 56, 64, 59, 61, 62, 58, 65, 60, 63, 57, 67, 68, 70, 72, 71, 74, 73, 75, 78, 76, 80, 79, 81, 82, 83

Offset: 0

Scott R. Shannon, Mar 09 2021

The sequence is finite due to the finite number of positive integers with distinct digits, see A010784, although the exact number of terms is currently unknown.

			a(1) = 1 as 1 has one distinct digit and a(0)+1 = 0+1 = 1 which has one distinct digit 0.
a(6) = 7 as 7 has one distinct digit and a(5)+7 = 5+7 = 12 which has two distinct digits. Note that 6 is the first skipped number as a(5)+6 = 5+6 = 11 has 1 as a duplicate digit.
a(11) = 13 as 13 has two distinct digits and a(10)+13 = 10+13 = 23 which has two distinct digits. Note that 11 and 12 are skipped as 11 has 1 as a duplicate digit while a(10)+12 = 10+12 = 22 has 2 as a duplicate digit.

Scott R. Shannon, Image of the first 60000 terms. The green line is a(n) = n.

Cf. A336285, A342382, A010784, A043537, A043096, A276633, A002378.

Block[{a = {0}, k, m = 10^4}, Do[k = 1; While[Nand[FreeQ[a, k], AllTrue[DigitCount[a[[-1]] + k], # < 2 &], AllTrue[DigitCount[k], # < 2 &]], If[k > m, Break[]]; k++]; If[k > m, Break[]]; AppendTo[a, k], {i, 76}]; a] (* Michael De Vlieger, Mar 11 2021 *)

def agen():
  alst, aset = [0], {0}
  yield 0
  while True:
    an = 1
    while True:
      while an in aset: an += 1
      stran, t = str(an), str(alst[-1] + an)
      if len(stran) == len(set(stran)) and len(t) == len(set(t)):
        alst.append(an); aset.add(an); yield an; break
      an += 1
g = agen()
print([next(g) for n in range(77)]) # Michael S. Branicky, Mar 11 2021

cp's OEIS Frontend

A067581 a(n) = smallest integer not yet in the sequence with no digits in common with a(n-1), a(0)=0.

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A342441 a(1) = 1; for n > 1, a(n) is the least positive integer not occurring earlier such that a(n-1)+a(n) shares no digit with either a(n-1) or a(n).

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A336285 a(0) = 0; for n > 0, a(n) is the least positive integer not occurring earlier such that the digits in a(n-1)+a(n) are all distinct.

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A338466 a(0) = 0; for n > 0, a(n) is the least positive integer not occurring earlier such that the digits in a(n-1)*a(n) are all distinct.

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A342442 a(1) = 2; for n > 1, a(n) is the least positive integer not occurring earlier such that a(n-1)*a(n) shares no digit with either a(n-1) or a(n).

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A239664 a(n) = 1, a(n+1) = smallest number not occurring earlier, having with a(n) neither a common digit nor a common divisor.

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A276512 a(n) = smallest integer not yet in the sequence with no digits in common with a(n-2); a(0)=0, a(1)=1.

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A342382 a(0) = 0; for n > 0, a(n) is the least positive integer not occurring earlier such that both the digits in a(n) and the digits in a(n-1)*a(n) are all distinct.

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