A052383 -id:A052383 - cp's OEIS frontend

A217489 Least positive integer without a digit 1, not listed earlier and not divisible by any digit of the preceding term.

2, 3, 4, 5, 6, 7, 8, 9, 20, 23, 25, 27, 29, 33, 22, 35, 26, 37, 32, 43, 34, 38, 28, 39, 40, 30, 44, 42, 45, 46, 47, 50, 24, 49, 53, 52, 57, 36, 55, 48, 54, 58, 59, 56, 62, 63, 64, 65, 67, 68, 69, 70, 60, 73, 74, 66, 75, 72, 79, 76, 80, 77, 78, 82, 83, 85, 84, 86, 87, 89, 92, 93, 88, 90

Offset: 1

M. F. Hasler and Eric Angelini, Oct 04 2012

This sequence contains all terms of A052383 that are not divisible by 2520. - Peter Kagey, Nov 04 2015
From Robert Israel, Jan 03 2016: (Start)
Here is a proof of Peter Kagey's comment:
Any number x in A052383 will eventually appear in the sequence if there are infinitely many members of the sequence containing no digit that divides x.
If k in A052383 is coprime to 210 (and thus not divisible by any digit > 1), then k is in the sequence.
The numbers 2...23 with number of 2's not divisible by 3, and 5...57 with number of 5's == 2,4 or 5 (mod 6) are coprime to 210, and thus are in the sequence.
The repunits k...k with k = 5 or 7 and an even number of digits are not divisible by 2 or 3, and thus they are in the sequence.
The repunits k...k with k = 2,3,4,6,8, or 9 and number of digits not divisible by 6 are not divisible by 5 or 7, and thus they are in the sequence. Any x in A052383 not divisible by 2520 is not divisible by one of the digits 2,3,...9, and thus is in the sequence. (End)

Peter Kagey, Table of n, a(n) for n = 1..10000
E. Angelini, (n+1) is not divisible by any digit of n, Oct 04 2012
E. Angelini, (n+1) is not divisible by any digit of n [Cached copy, with permission]

Cf. A002275, A038603, A052383.

Sequence A217491 is a variant of the same idea (where injectivity is strengthened to strict monotonicity).

N:= 1000: # to get all terms before the first that exceeds N
A[1]:= 2:
Av:= remove(t -> has(convert(t,base,10),1),{$3..N}):
for n from 2 do
  d:= convert(convert(A[n-1],base,10),set) minus {0};
  Ad:= remove(t -> ormap(y -> t mod y = 0, d) , Av);
  if nops(Ad) = 0 then break fi;
  A[n]:= min(Ad);
  Av:= Av minus {A[n]};
od:
seq(A[i],i=1..n-1); # Robert Israel, Jan 03 2016

a = {2}; Do[k = 1; While[Or[First@ DigitCount@ k > 0, MemberQ[a, k], Total[Boole@ Divisible[k, #] & /@ (IntegerDigits@ a[[n - 1]] /. 0 -> Nothing)] > 0], k++]; AppendTo[a, k], {n, 2, 74}]; a (* Michael De Vlieger, Nov 05 2015 *)

A255357 Natural numbers n, other than multiples of 10, such that n, n^2 and n^3 lack the digit 1 in their decimal expansion.

Original entry on oeis.org

2, 3, 7, 62, 63, 65, 66, 67, 74, 76, 77, 78, 84, 86, 87, 92, 93, 94, 95, 202, 207, 274, 275, 282, 284, 287, 288, 292, 295, 298, 305, 307, 452, 453, 457, 587, 588, 592, 594, 607, 624, 632, 635, 636, 637, 638, 653, 664, 665, 666, 667, 668, 675, 686, 688, 695, 697, 698, 702, 703, 705, 707

Offset: 1

Zak Seidov, Feb 23 2015

			Numbers {2, 3, 7, 62, 63}, their squares {4, 9, 49, 3844, 3969} and cubes {8, 27, 343, 238328, 250047} all are "one-less".

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

Cf. A000027, A052383, A067251, A255398, A255430, A255431.

filter:= proc(n) local L,i;
  for i from 1 to 3 do
    L:= convert(n^i,base,10);
    if member(1,L) then return false fi;
  od;
  true
end proc:
select(filter, [seq(seq(10*i+j, j=2..8),i=0..100)]); # Robert Israel, Apr 03 2024

Select[Range[800],NumberDigit[#,0]!=0&&FreeQ[Flatten[ IntegerDigits/@ {#,#^2,#^3}],1]&] (* Harvey P. Dale, Sep 24 2021 *)

Definition modified by Harvey P. Dale, Sep 24 2021

A275533 Least positive number m such that m^k for k = 1..n have no digit 1.

Original entry on oeis.org

2, 2, 2, 74, 94, 305, 2975

Offset: 1

Zak Seidov, Jul 31 2016

It appears that there are no more terms in this sequence.
If they exist, a(8) > 127231981 and a(9) > 107895940. - J.W.L. (Jan) Eerland, Nov 26 2022
If they exist, a(8)-a(15) > 10^13 and a(16) > 10^32; a(17) and beyond do not exist. - Michael S. Branicky, Apr 05 2024

			2975^k, k = 1..7 = 2975, 8850625, 26330609375, 78333562890625, 233042349599609375, 693300990058837890625, 2062570445425042724609375 (all are oneless), while 2975^8 = 6136147075139502105712890625 has 5 ones.

Cf. A052383 (numbers without 1 as a digit), A371706.

Table[{n,m=1;Monitor[Parallelize[While[True,If[Length[DeleteCases[Table[If[MemberQ[IntegerDigits[m^k],1]==False,k,a],{k,1,n}],a]]==n,Break[]];m++];m],m]},{n,1,7}] (* J.W.L. (Jan) Eerland, Nov 26 2022 *)

from itertools import count
def A275533(n): return next(m for m in count(1) if not any('1' in str(m**k) for k in range(1,n+1))) # Chai Wah Wu, Apr 04 2024

a(n) <= a(n+1). - Michael S. Branicky, Apr 05 2024

A387032 Numbers k with digits different from 0 and 1.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 22, 23, 24, 25, 26, 27, 28, 29, 32, 33, 34, 35, 36, 37, 38, 39, 42, 43, 44, 45, 46, 47, 48, 49, 52, 53, 54, 55, 56, 57, 58, 59, 62, 63, 64, 65, 66, 67, 68, 69, 72, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 88, 89, 92, 93, 94, 95, 96, 97, 98, 99, 222

Offset: 1

David A. Corneth, Aug 13 2025

A062998 contains numbers like 123, 124, 125,.. which are not in this sequence. - R. J. Mathar, Aug 14 2025

A037344 contains numbers like 2047 and 4095 which are not in this sequence. - R. J. Mathar, Aug 14 2025

			2 is in the sequence since it does not contain 0 nor 1.
12 is not in the sequence since it has digit 1.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Intersection of A052382 and A052383.

Cf. A037344, A172424.

isA387032 := proc(n)
    local d ;
    for d in convert(n,base,10) do
        if d <=1 then
            return false;
        end if;
    end do:
    true ;
end proc:
A387032 := proc(n)
    option remember ;
    local a;
    if n = 1 then
        2;
    else
        for a from procname(n-1)+1 do
            if isA387032(a) then
                return a;
            end if;
        end do;
    end if;
end proc:
seq(A387032(n),n=1..200) ; # R. J. Mathar, Aug 14 2025

Select[Range[222], Total@ DigitCount[#, 10, {0, 1}] == 0 &] (* Michael De Vlieger, Aug 13 2025 *)

is(n) = if(n <= 0, return(0)); Set(digits(n))[1] >= 2

def ok(n): return {"0","1"} & set(str(n)) == set()
print([k for k in range(223) if ok(k)]) # Michael S. Branicky, Aug 13 2025

def A387032(n):
    m = ((k:=7*n+1).bit_length()-1)//3
    return sum((2+((k-(1<<3*m))//(7<<3*j)&7))*10**j for j in range(m)) # Chai Wah Wu, Aug 13 2025

A255431 Numbers n such that integers n through n+6 and their squares all lack the digit 1 in their decimal expansion.

Original entry on oeis.org

22, 62, 82, 472, 522, 662, 822, 832, 932, 972, 2362, 2382, 2522, 2562, 2632, 2972, 4522, 4832, 4862, 4882, 4932, 4972, 5022, 5062, 5082, 5322, 5472, 6062, 6382, 6632, 6662, 6782, 6972, 7022, 7382, 7432, 7472, 7632, 8322, 8332, 8362, 8672, 8882

Offset: 1

Zak Seidov, Feb 23 2015

Integers n through n+6 are terms in A255430.

All terms are congruent to 2 mod 10.

			22,23,24,25,26,27,28 and their squares 484,529,576,625,676,729,784 are "one-less".

Cf. A000027, A052383, A067251, A255398, A255430.

A275512 The consecutive sizes of the blocks of terms containing at least one digit "1" are given by the sequence itself.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 20, 12, 13, 14, 22, 15, 16, 17, 18, 23, 19, 21, 31, 41, 51, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 71, 81, 91, 100, 101, 62, 63, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84, 85, 86, 87, 88, 89, 90

Offset: 1

Eric Angelini, Jul 31 2016

The sequence starts with a(1) = 1 and is extended with the smallest integer not yet used that does not lead to a contradiction.

			The blocks of terms including at least a digit "1" are indicated here by parentheses; the successive block-sizes are 1, 2, 3, 4, 5,... which reproduces the sequence itself: (1),2,3,4,5,6,7,8,9,(10,11),20,(12,13,14),22,(15,16,17,18),23,(19,21,31,41,51),24...

Cf. A011531, A052383.

isA011531 := proc(n)
    nops(convert(convert(n,base,10),set) intersect {1}) > 0 ;
    simplify(%) ;
end proc:
A011531_next := proc(n)
    local a;
    for a from n+1 do
        if isA011531(a) then
            return a;
        end if;
    end do:
end proc:
isA052383 := proc(n)
    not isA011531(n) ;
end proc:
A052383_next := proc(n)
    local a;
    for a from n+1 do
        if isA052383(a) then
            return a;
        end if;
    end do:
end proc:
A275512grp1 := proc(a)
    local idx1,n ;
    idx1 := 0 ;
    if isA052383(a) then
        return 0;
    end if;
    for n from 1 do
        if isA011531(A275512(n)) then
            if n =1 then
                idx1 := 1;
            elif isA052383(A275512(n-1)) then
                idx1 := idx1+1 ;
            end if;
        end if;
        if A275512(n) = a then
            return idx1 ;
        end if;
    end do:
end proc:
A275512 := proc(n)
    option remember;
    local a,aprev,prev1,grp1,d,seen,reqlen,npr;
    if n =1 then
        1;
    else
        for a from 2 do
            seen := false;
            for npr from 1 to n-1 do
                if procname(npr) = a then
                    seen := true;
                    break;
                end if;
            end do:
            if not seen then
                aprev := procname(n-1) ;
                if isA052383(aprev) then
                    return a;
                else
                    prev1 := 0 ;
                    for d from 1 to n-1 do
                        if isA011531(procname(n-d)) then
                            prev1 := prev1+1 ;
                        else
                            break;
                        end if;
                    end do:
                    grp1 := A275512grp1(aprev) ;
                    reqlen := procname(grp1) ;
                    if reqlen > prev1 then
                        return A011531_next(aprev) ;
                    elif n-prev1-1 > 0 then
                        return A052383_next(procname(n-prev1-1)) ;
                    else
                        return 2 ;
                    end if;
                end if;
            end if;
        end do;
    end if;
end proc:
seq(A275512(n),n=1..100) ; # R. J. Mathar, Jul 31 2016

A288538 Lexicographically earliest sequence of distinct positive terms such that, for any n > 0, n * a(n) does not contain the digit 1 in decimal base.

Original entry on oeis.org

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

Offset: 1

Rémy Sigrist, Jun 24 2017

This sequence is a permutation of the natural numbers; for any n > 0, there are infinitely many values v such that n * v does not contain the digit 1 in decimal base (for example: 2 * A004290(n) / n * 10^k for k = 0, 1, 2,...), hence n will eventually be chosen.
This sequence is self-inverse.
If n is a fixed point, then n^2 belongs to A052383.
The first fixed points are: 3, 6, 7, 8, 15, 20, 23, 24, 25, 26, 27, 28, 45, 47, 60, 64, 65, 66, 67, 68, 75, 76, 77, 78, ...

			The first terms, alongside n * a(n), are:
n       a(n)    n * a(n)
--      ----    --------
1       2       2
2       1       2
3       3       9
4       5       20
5       4       20
6       6       36
7       7       49
8       8       64
9       10      90
10      9       90
11      19      209
12      17      204
13      16      208
14      18      252
15      15      225
16      13      208
17      12      204
18      14      252
19      11      209
20      20      400

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A288538
Index entries for sequences that are permutations of the natural numbers
Rémy Sigrist, Logarithmic scatterplot of the first 10000 terms of n * a(n)

Cf. A004290, A052383.

A362064 Lexicographically earliest sequence of distinct positive integers such that the digit "1" is neither present in a(n) nor in a(n) + a(n+1).

Original entry on oeis.org

2, 3, 4, 5, 20, 6, 22, 7, 23, 9, 24, 8, 25, 27, 26, 28, 29, 30, 32, 33, 34, 35, 37, 36, 38, 39, 40, 42, 43, 44, 45, 47, 46, 48, 49, 50, 200, 52, 202, 53, 203, 54, 204, 55, 205, 57, 206, 56, 207, 58, 208, 59, 209, 60, 220, 62, 222, 63, 223, 64, 224, 65, 225, 67

Offset: 1

Eric Angelini and Cécile Angelini, Apr 07 2023

			2 + 3 = 5; 3 + 4 = 7; 4 + 5 = 9; but as 5 + 6 = 11 we cannot use 6 nor 7 (sum 12), 8 (sum 13) and 9 (sum 14); we cannot use the integers 10 to 19 (as a 1 is present in them), so a(5) = 20 as 5 + 20 = 25, etc.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A052383, A299957.

from itertools import islice
def jump1(n):
    s = str(n)
    return n if "1" not in s else int(s[:(i:=s.index("1"))]+"2"+"0"*(len(s)-i-1))
def agen(): # generator of terms
    an, aset, mink = 2, {2}, 3
    while True:
        yield an
        k = mink
        while k in aset or "1" in str(an+k): k = jump1(k+1)
        an = k
        aset.add(an)
        while mink in aset: mink = jump1(mink+1)
print(list(islice(agen(), 64))) # Michael S. Branicky, Apr 07 2023

a(27) and beyond from Michael S. Branicky, Apr 07 2023

A338287 Decimal expansion of the sum of reciprocals of the numbers that are not pandigital numbers (version 2, A171102).

Original entry on oeis.org

6, 5, 7, 4, 3, 3, 1, 1, 1, 0, 1, 8, 5, 3, 2, 8, 1, 9, 6, 7, 3, 4, 5, 8, 3, 1, 6, 7, 6, 8, 0, 8, 6, 8, 4, 1, 1, 6, 8, 5, 3, 4, 4, 1, 0, 6, 6, 3, 5, 3, 9, 8, 1, 6, 1, 0, 5, 0, 4, 3, 9, 2, 6, 3, 4, 6, 1, 3, 8, 7, 3, 8, 7, 3, 7, 1, 8, 5, 2, 6, 8, 0, 3, 4, 7, 8, 2

Offset: 2

Amiram Eldar, Oct 20 2020

The sum of the reciprocals of the terms of the complement of A171102: numbers with at most 9 distinct digits. It is the union of the 10 sequences of numbers without a single given digit (see the Crossrefs section).

The terms in the data section were taken from the 200 decimal digits given by Strich and Müller (2020).

			65.74331110185328196734583167680868411685344106635398...

Robert Strich and Eric Müller, Ziffernreduzierte Zahlen, in: E. Specht, E. Quaisser and P. Bauermann (eds.), 50 Jahre Bundeswettbewerb Mathematik, Springer Spektrum, Berlin, Heidelberg, 2020, pp. 171-182.

Cf. A171102, A179954, A082830, A082831, A082832, A082833, A082834, A082835, A082836, A082837, A082838, A082839.

Cf. A052382 (numbers without the digit 0), A052383 (without 1), A052404 (without 2), A052405 (without 3), A052406 (without 4), A052413 (without 5), A052414 (without 6), A052419 (without 7), A052421 (without 8), A007095 (without 9).

Equals 1/1 + 1/2 + 1/3 + ... + 1/1023456788 + 1/1023456790 + ..., i.e., A171102(1) = 1023456789 is the first number whose reciprocal is not in the sum.

A359534 Numbers that don’t have a 1 and yet have one.

Original entry on oeis.org

2000008, 2000080, 2000082, 2000083, 2000084, 2000085, 2000086, 2000087, 2000088, 2000089, 2000800, 2000802, 2000803, 2000804, 2000805, 2000806, 2000807, 2000808, 2000809, 2000820, 2000822, 2000823, 2000824, 2000825, 2000826, 2000827, 2000828, 2000829, 2000830, 2000832, 2000833, 2000834, 2000835, 2000836, 2000837, 2000838

Offset: 1

Eric Angelini, Oct 15 2023

			The first three terms are 2000008, 2000080 and 2000082 where no digit 1 is present, and yet we can read the substring ONE when those numbers are written in English: two milli(ON E)ight, two milli(ON E)ighty, two milli(ON E)ighty two, etc.
We cannot use Eleven after milliON, nor Eighteen, as this would produce at least one digit 1.

Cf. A052383, A345711.

Select[Range[#, # + 1000] &[2*10^6], And[StringContainsQ[StringJoin@ DeleteCases[Characters@ IntegerName[#, "Words"], ?(! LetterQ[#] &)], "one"], DigitCount[#, 10, 1] == 0] &] (* _Michael De Vlieger, Oct 15 2023 *)

from num2words import num2words
def n2w(n): return num2words(n).replace(" and","").replace(chr(44),"").replace("-","").replace(" ","")
def ok(n): return "1" not in str(n) and "one" in n2w(n)
print([k for k in range(2000840) if ok(k)]) # Michael S. Branicky, Oct 15 2023

cp's OEIS Frontend

A217489 Least positive integer without a digit 1, not listed earlier and not divisible by any digit of the preceding term.

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A255357 Natural numbers n, other than multiples of 10, such that n, n^2 and n^3 lack the digit 1 in their decimal expansion.

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Extensions

A275533 Least positive number m such that m^k for k = 1..n have no digit 1.

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Formula

A387032 Numbers k with digits different from 0 and 1.

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Python

A255431 Numbers n such that integers n through n+6 and their squares all lack the digit 1 in their decimal expansion.

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A275512 The consecutive sizes of the blocks of terms containing at least one digit "1" are given by the sequence itself.

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Maple

A288538 Lexicographically earliest sequence of distinct positive terms such that, for any n > 0, n * a(n) does not contain the digit 1 in decimal base.

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A362064 Lexicographically earliest sequence of distinct positive integers such that the digit "1" is neither present in a(n) nor in a(n) + a(n+1).

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Python