A038603 -id:A038603 - 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 *)

A383749 Positive numbers k whose decimal expansion does not contain the decimal expansion of any proper divisor of k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 23, 27, 29, 34, 37, 38, 43, 46, 47, 49, 53, 54, 56, 57, 58, 59, 67, 68, 69, 73, 74, 76, 78, 79, 83, 86, 87, 89, 94, 97, 98, 203, 207, 209, 223, 227, 229, 233, 239, 247, 249, 253, 257, 259, 263, 267, 269, 277, 283, 289, 293, 299, 307

Offset: 1

Rémy Sigrist, May 08 2025

Also fixed points of A121042.
This sequence is infinite as it contains A173041.
a(n) > 5 contains no decimal digit 1 and does not end in 2 or 5. - Michael S. Branicky, May 11 2025

			The proper divisors of 54 are 1, 2, 3, 6, 9, 18 and 27; none of them appear in the decimal expansion of 54 so 54 belongs to this sequence.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A011531, A027751, A038772, A121042, A383592.

A038603 and A173041 are subsequences.

A383749Q[k_] := SelectFirst[Divisors[k], StringContainsQ[IntegerString[k], IntegerString[#]] &] == k;
Select[Range[500], A383749Q] (* Paolo Xausa, May 12 2025 *)

is(n, base = 10) = {
    my (d = digits(n, base));
    for (i = 1, #d,
        if (d[i],
            for (j = i, #d,
                if ((i!=1 || j!=#d) && n % fromdigits(d[i..j], base)==0,
                    return (0);););););
    return (1);}

from itertools import count, islice
from sympy import divisors
def A383749_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:not any(dA383749_list = list(islice(A383749_gen(),40)) # Chai Wah Wu, May 10 2025

def ok(n):
    s = str(n)
    subs = (s[i:j] for i in range(len(s)) for j in range(i+1, len(s)+1) if s[i]!='0')
    return n and not any(n%v == 0 for ss in subs if n > (v:=int(ss)) > 0)
print([k for k in range(308) if ok(k)]) # Michael S. Branicky, May 11 2025

A217491 Next largest positive integer without a digit 1 and not divisible by any digit of the preceding term.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 20, 23, 25, 27, 29, 33, 34, 35, 37, 38, 43, 46, 47, 50, 52, 53, 56, 57, 58, 59, 62, 63, 64, 65, 67, 68, 69, 70, 72, 73, 74, 75, 76, 79, 80, 82, 83, 85, 86, 87, 89, 92, 93, 94, 95, 96, 97, 200, 203, 205, 207, 209, 223, 227, 229, 233, 235, 239, 245, 247, 249

Offset: 1

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

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]

A variant of A217489.

Cf. A038603. - Charles R Greathouse IV, Oct 04 2012

A307636 Numbers k with property that no two divisors of k share a common digit.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 23, 27, 29, 37, 43, 47, 49, 53, 59, 67, 73, 79, 83, 86, 87, 89, 97, 223, 227, 229, 233, 239, 257, 263, 267, 269, 277, 283, 293, 307, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 409, 433, 439, 443, 449, 457, 463, 467, 479, 487, 499, 503

Offset: 1

Giorgos Kalogeropoulos, May 03 2019

			9566 is such a number because its divisors are  1, 2, 4783 and 9566, and no two of them share the same digit.

Robert Israel, Table of n, a(n) for n = 1..10000
Giorgos Kalogeropoulos, Hostile Divisor Numbers, Code Golf, May 2019.

A038603 is a subsequence.

filter:= proc(n) local D,i,j;
  D:= map(t -> convert(convert(t,base,10),set), convert(numtheory:-divisors(n),list));
  for i from 2 to nops(D) do
    for j from 1 to i-1 do
       if D[i] intersect D[j] <> {} then return false fi
  od od;
  true
end proc:
select(filter, [$1..1000]); # Robert Israel, Jul 07 2019

Select[Range@1000,!Or@@IntersectingQ@@@Subsets[IntegerDigits@Divisors[#],{2}]&]

isok(k) = {my(d = divisors(k), dd = apply(x->Set(digits(x)), d)); for (i=1, #dd, for (j=i+1, #dd, if (#setintersect(dd[i], dd[j]), return (0)););); return (1);} \\ Michel Marcus, Jul 07 2019

from itertools import count, combinations, islice
from sympy import divisors
def A307636gen(): return filter(lambda n:all(len(set(s[0])&set(s[1])) == 0 for s in combinations((str(d) for d in divisors(n,generator=True)),2)),count(1))
A307636_list = list(islice(A307636gen(),20)) # Chai Wah Wu, Dec 08 2021

A386328 Primes without {1, 2} as digits.

Original entry on oeis.org

3, 5, 7, 37, 43, 47, 53, 59, 67, 73, 79, 83, 89, 97, 307, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 409, 433, 439, 443, 449, 457, 463, 467, 479, 487, 499, 503, 509, 547, 557, 563, 569, 577, 587, 593, 599, 607, 643, 647, 653, 659, 673, 677, 683, 709

Offset: 1

Jason Bard, Jul 19 2025

Intersection of A038603 and A038604.

Cf. A000040, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [0, 3, 4, 5, 6, 7, 8, 9]];

Select[Prime[Range[120]], DigitCount[#, 10, 1] == 0 && DigitCount[#, 10, 2] == 0 &]

primes_with(, 1, [0, 3, 4, 5, 6, 7, 8, 9]) \\ uses function in A385776

print(list(islice(primes_with("03456789"), 41))) # uses function/imports in A385776

A386329 Primes without {1, 3} as digits.

Original entry on oeis.org

2, 5, 7, 29, 47, 59, 67, 79, 89, 97, 227, 229, 257, 269, 277, 409, 449, 457, 467, 479, 487, 499, 509, 547, 557, 569, 577, 587, 599, 607, 647, 659, 677, 709, 727, 757, 769, 787, 797, 809, 827, 829, 857, 859, 877, 887, 907, 929, 947, 967, 977, 997, 2027, 2029, 2069

Offset: 1

Jason Bard, Jul 19 2025

Intersection of A038603 and A038611.

Cf. A000040, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [0, 2, 4, 5, 6, 7, 8, 9]];

Select[Prime[Range[120]], DigitCount[#, 10, 1] == 0 && DigitCount[#, 10, 3] == 0 &]

primes_with(, 1, [0, 2, 4, 5, 6, 7, 8, 9]) \\ uses function in A385776

print(list(islice(primes_with("02456789"), 41))) # uses function/imports in A385776

A386330 Primes without {1, 4} as digits.

Original entry on oeis.org

2, 3, 5, 7, 23, 29, 37, 53, 59, 67, 73, 79, 83, 89, 97, 223, 227, 229, 233, 239, 257, 263, 269, 277, 283, 293, 307, 337, 353, 359, 367, 373, 379, 383, 389, 397, 503, 509, 523, 557, 563, 569, 577, 587, 593, 599, 607, 653, 659, 673, 677, 683, 709, 727, 733, 739

Offset: 1

Jason Bard, Jul 19 2025

Intersection of A038603 and A038612.

Cf. A000040, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [0, 2, 3, 5, 6, 7, 8, 9]];

Select[Prime[Range[120]], DigitCount[#, 10, 1] == 0 && DigitCount[#, 10, 4] == 0 &]

primes_with(, 1, [0, 2, 3, 5, 6, 7, 8, 9]) \\ uses function in A385776

print(list(islice(primes_with("02356789"), 41))) # uses function/imports in A385776

A386331 Primes without {1, 5} as digits.

Original entry on oeis.org

2, 3, 7, 23, 29, 37, 43, 47, 67, 73, 79, 83, 89, 97, 223, 227, 229, 233, 239, 263, 269, 277, 283, 293, 307, 337, 347, 349, 367, 373, 379, 383, 389, 397, 409, 433, 439, 443, 449, 463, 467, 479, 487, 499, 607, 643, 647, 673, 677, 683, 709, 727, 733, 739, 743, 769

Offset: 1

Jason Bard, Jul 19 2025

Intersection of A038603 and A038613.

Cf. A000040, A020453, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [0, 2, 3, 4, 6, 7, 8, 9]];

f:= n-> (l-> add([0, $2..4, $6..9][l[j]+1]*10^(j-1), j=1..nops(l)))(convert(n, base, 8)):
select(isprime, [seq(f(i), i=0..600)])[];  # Alois P. Heinz, Jul 19 2025

Select[Prime[Range[120]], DigitCount[#, 10, 1] == 0 && DigitCount[#, 10, 5] == 0 &]

primes_with(, 1, [0, 2, 3, 4, 6, 7, 8, 9]) \\ uses function in A385776

print(list(islice(primes_with("02346789"), 41))) # uses function/imports in A385776

A386332 Primes without {1, 6} as digits.

Original entry on oeis.org

2, 3, 5, 7, 23, 29, 37, 43, 47, 53, 59, 73, 79, 83, 89, 97, 223, 227, 229, 233, 239, 257, 277, 283, 293, 307, 337, 347, 349, 353, 359, 373, 379, 383, 389, 397, 409, 433, 439, 443, 449, 457, 479, 487, 499, 503, 509, 523, 547, 557, 577, 587, 593, 599, 709, 727

Offset: 1

Jason Bard, Jul 19 2025

Intersection of A038603 and A038614.

Cf. A000040, A020454, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [0, 2, 3, 4, 5, 7, 8, 9]];

Select[Prime[Range[120]], DigitCount[#, 10, 1] == 0 && DigitCount[#, 10, 6] == 0 &]

primes_with(, 1, [0, 2, 3, 4, 5, 7, 8, 9]) \\ uses function in A385776

print(list(islice(primes_with("02345789"), 41))) # uses function/imports in A385776

A386333 Primes without {1, 7} as digits.

Original entry on oeis.org

2, 3, 5, 23, 29, 43, 53, 59, 83, 89, 223, 229, 233, 239, 263, 269, 283, 293, 349, 353, 359, 383, 389, 409, 433, 439, 443, 449, 463, 499, 503, 509, 523, 563, 569, 593, 599, 643, 653, 659, 683, 809, 823, 829, 839, 853, 859, 863, 883, 929, 953, 983, 2003, 2029, 2039

Offset: 1

Jason Bard, Jul 19 2025

Intersection of A038603 and A038615.

Cf. A000040, A020455, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [0, 2, 3, 4, 5, 6, 8, 9]];

Select[Prime[Range[120]], DigitCount[#, 10, 1] == 0 && DigitCount[#, 10, 7] == 0 &]

primes_with(, 1, [0, 2, 3, 4, 5, 6, 8, 9]) \\ uses function in A385776

print(list(islice(primes_with("02345689"), 41))) # uses function/imports in A385776

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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Maple

Mathematica

A383749 Positive numbers k whose decimal expansion does not contain the decimal expansion of any proper divisor of k.

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Mathematica

PARI

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Python

A217491 Next largest positive integer without a digit 1 and not divisible by any digit of the preceding term.

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A307636 Numbers k with property that no two divisors of k share a common digit.

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Examples

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Maple

Mathematica

PARI

Python

A386328 Primes without {1, 2} as digits.

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Magma

Mathematica

PARI

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A386329 Primes without {1, 3} as digits.

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Magma

Mathematica

PARI

Python

A386330 Primes without {1, 4} as digits.

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Magma

Mathematica

PARI

Python

A386331 Primes without {1, 5} as digits.

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