A038772 -id:A038772 - cp's OEIS frontend

A082943 Positive numbers not divisible by any of their digits nor by the sum of their digits.

23, 29, 34, 37, 38, 43, 46, 47, 49, 53, 56, 57, 58, 59, 67, 68, 69, 73, 74, 76, 78, 79, 83, 86, 87, 89, 94, 97, 98, 223, 227, 229, 233, 239, 249, 253, 257, 259, 263, 267, 269, 277, 283, 289, 293, 299, 323, 329, 334, 337, 338, 343, 346, 347, 349, 353, 356

Offset: 1

Meenakshi Srikanth (menakan_s(AT)yahoo.com), Apr 30 2003

The definition implies that no digit is zero. - N. J. A. Sloane, Mar 27 2025

			38 is neither divisible by 3 nor 8 nor 11 (i.e. 3+8).

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A038772, A382239.

filter:= proc(n) local L;
  L:= convert(n,base,10);
  not member(0,L) and not ormap(t -> n mod t = 0, [op(L),convert(L,`+`)])
end proc:
select(filter, [$1..1000]); # Robert Israel, Mar 19 2025

test1[n_] := Module[{dig = IntegerDigits[n]}, Union[Table[IntegerQ[n/dig[[i]]], {i, Length[dig]}]]] == {False};
test2[n_] := Module[{dig = IntegerDigits[n]}, Not[IntegerQ[n/Sum[dig[[i]], {i, Length[dig]}]]]];
Table[If[test1[n] && test2[n], n, 0], {n, 200}] // Union // Rest (* José María Grau Ribas, Feb 17 2010 *)
ndQ[n_]:=Module[{idn=IntegerDigits[n]},FreeQ[idn,0]&&NoneTrue[n/ Join[ idn, {Total[idn]}],IntegerQ]]; Select[Range[2000],ndQ] (* Harvey P. Dale, Oct 19 2016 *)

More terms from Harvey P. Dale, Oct 19 2016

A352380 Numbers k such that no nonzero digit of 3k divides 3k.

Original entry on oeis.org

9, 18, 19, 23, 26, 29, 69, 83, 89, 143, 149, 158, 159, 163, 166, 169, 186, 193, 196, 199, 203, 209, 219, 223, 229, 233, 236, 249, 253, 258, 260, 263, 269, 283, 286, 289, 290, 293, 298, 299, 319, 323, 326, 669, 683, 689, 743, 759, 763, 803, 809, 823, 829, 833, 849, 853, 859, 863, 869, 883, 893, 899

Offset: 1

Eric Angelini and Carole Dubois, Mar 14 2022

			a(1) = 9 and 3*9 = 27 is not divisible by 2 or 7;
a(2) = 18 and 3*18 = 54 is not divisible by 5 or 4;
a(3) = 19 and 3*19 = 57 is not divisible by 5 or 7;
a(4) = 23 and 3*23 = 69 is not divisible by 6 or 9; etc.
31 is not in the sequence as 3*31 = 93 is divisible by 3.

Cf. A038772, A352379, A352381, A352382.

q[n_] := AllTrue[IntegerDigits[3*n], # == 0 || !Divisible[3*n, #] &]; Select[Range[900], q] (* Amiram Eldar, Mar 14 2022 *)

def ok(n): return not any(3*n%int(d)==0 for d in set(str(3*n)) if d!='0')
print([k for k in range(1, 900) if ok(k)]) # Michael S. Branicky, Mar 14 2022

A352381 Numbers k such that no nonzero digit of 4k divides 4k.

Original entry on oeis.org

14, 17, 19, 77, 89, 94, 95, 97, 127, 134, 139, 147, 149, 164, 167, 169, 177, 184, 190, 194, 195, 197, 199, 209, 215, 217, 227, 239, 244, 245, 247, 764, 767, 769, 827, 839, 844, 845, 847, 877, 884, 889, 899, 914, 917, 919, 925, 934, 940, 944, 947, 949, 959, 965, 967, 977, 989, 995, 997, 1259, 1264, 1267

Offset: 1

Eric Angelini and Carole Dubois, Mar 14 2022

			a(1) = 14 and 4*14 = 56 is not divisible by 5 or 6;
a(2) = 17 and 4*17 = 68 is not divisible by 6 or 8;
a(3) = 19 and 4*19 = 76 is not divisible by 7 or 6;
a(4) = 77 and 4*77 = 308 is not divisible by 3, 0 or 8; etc.
26 is not in the sequence as 4*26 = 104 is divisible by 1 (and 4).

Cf. A038772, A352379, A352380, A352382.

q[n_] := AllTrue[IntegerDigits[4*n], # == 0 || !Divisible[4*n, #] &]; Select[Range[1270], q] (* Amiram Eldar, Mar 14 2022 *)

def ok(n): return not any(4*n%int(d)==0 for d in set(str(4*n)) if d!='0')
print([k for k in range(1, 978) if ok(k)]) # Michael S. Branicky, Mar 14 2022

A352382 Numbers k such that no nonzero digit of 5k divides 5k.

Original entry on oeis.org

74, 76, 86, 94, 98, 134, 146, 152, 156, 158, 166, 172, 174, 178, 194, 196, 614, 674, 676, 686, 694, 698, 734, 740, 746, 752, 754, 758, 766, 772, 778, 794, 796, 806, 814, 818, 866, 874, 878, 886, 894, 898, 926, 934, 938, 946, 954, 958, 974, 978, 986, 998, 1214, 1276, 1286, 1294, 1298, 1334, 1340

Offset: 1

Eric Angelini and Carole Dubois, Mar 14 2022

Only even terms (see the last line of the Example section to understand why).

			a(1) = 74 and 5*74 = 370 is not divisible by 3, 7 or 0;
a(2) = 76 and 5*76 = 380 is not divisible by 3, 8 or 0;
a(3) = 86 and 5*86 = 430 is not divisible by 4, 3 or 0;
a(4) = 94 and 5*94 = 470 is not divisible by 4, 7 or 0; etc.
93 is not in the sequence as 5*93 = 465 is divisible by 5.

Cf. A038772, A352379, A352380, A352381.

q[n_] := AllTrue[IntegerDigits[5*n], # == 0 || !Divisible[5*n, #] &]; Select[Range[1340], q] (* Amiram Eldar, Mar 14 2022 *)

def ok(n): return not any(5*n%int(d)==0 for d in set(str(5*n)) if d!='0')
print([k for k in range(1, 1277) if ok(k)]) # Michael S. Branicky, Mar 14 2022

A382239 Numbers not divisible by any of their digits nor by the sum of their digits. Digit 0 is allowed (and does not divide anything).

Original entry on oeis.org

23, 29, 34, 37, 38, 43, 46, 47, 49, 53, 56, 57, 58, 59, 67, 68, 69, 73, 74, 76, 78, 79, 83, 86, 87, 89, 94, 97, 98, 203, 223, 227, 229, 233, 239, 249, 253, 257, 259, 263, 267, 269, 277, 283, 289, 293, 299, 307, 323, 329, 334, 337, 338, 343, 346, 347, 349, 353, 356, 358, 359, 367, 373, 374, 376

Offset: 1

Robert Israel, Mar 19 2025

From a suggestion by Sergio Pimentel.

			a(5) = 38 is included because 38 is not divisible by 3, 8 or 3 + 8 = 11.
a(30) = 203 is included because 203 is not divisible by 2, 0, 3 or 2 + 0 + 3 = 5.

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

Subsequence of A052383.

Cf. A038772, A082943, A382237.

filter:= proc(n) local L;
  L:= subs(0=NULL,convert(n,base,10));
  not ormap(t -> n mod t = 0, [op(L),convert(L,`+`)])
end proc:
select(filter, [$1..1000]);

s= {};Do[t=Select[IntegerDigits[n],#>0&];AppendTo[t,Total[t]];If[NoneTrue[t,Mod[n,#]==00&],AppendTo[s,n]],{n,376}];s (* James C. McMahon, Mar 21 2025 *)

def ok(n):
    d = list(map(int, str(n)))
    return (s:=sum(d)) and n%s!=0 and all(n%di!=0 for di in set(d)-{0})
print([k for k in range(1, 377) if ok(k)]) # Michael S. Branicky, Apr 01 2025

n^k << a(n) < 2^n for n > 5 where k = log(10)/log(9). - Charles R Greathouse IV, Mar 20 2025

A244356 Numbers n such that n and n+1 are not divisible by any of their nonzero digits.

Original entry on oeis.org

37, 46, 53, 56, 57, 58, 67, 68, 73, 78, 86, 97, 307, 337, 346, 358, 373, 376, 379, 388, 397, 406, 429, 433, 446, 457, 466, 469, 473, 477, 478, 489, 493, 498, 506, 507, 508, 538, 553, 556, 557, 558, 577, 578, 586, 587, 588, 596, 597, 598, 646, 656, 657, 658, 667, 668, 669

Offset: 1

Derek Orr, Jun 26 2014

This is a subsequence of A038772.

All numbers end in a 3, 6, 7, 8, or 9.

			37 is not divisible by 3 or 7 and 38 is not divisible by 3 or 8. Thus 37 is a member of this sequence.

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

Cf. A038772, A237766.

filter:= proc(n) local L;
  L:= convert(convert(n,base,10), set) minus {0};
  not ormap(t -> n mod t = 0, L)
end proc:
B:= select(filter, {$1..1000}):
sort(convert(B intersect map(`-`,B,1), list)); # Robert Israel, Dec 08 2019

def a(n):
  for i in range(10**3):
    tot = 0
    for k in range(i,i+n):
      c = 0
      for b in str(k):
        if b != '0':
          if k%int(b)!=0:
            c += 1
      if c == len(str(k))-str(k).count('0'):
        tot += 1
    if tot == n:
      print(i,end=', ')
a(2)

A244357 Numbers n such that n, n+1, and n+2 are not divisible by any of their nonzero digits.

Original entry on oeis.org

56, 57, 67, 477, 506, 507, 556, 557, 577, 586, 587, 596, 597, 656, 657, 667, 668, 697, 757, 758, 778, 787, 788, 857, 858, 866, 867, 868, 877, 897, 956, 957, 976, 977, 978, 4077, 4097, 4457, 4477, 4497, 4657, 4677, 4757, 4857, 4897, 4997, 5056, 5057, 5066, 5067, 5077, 5096

Offset: 1

Derek Orr, Jun 26 2014

This is a subsequence of A244356.

All numbers end in a 6, 7, or 8.

			56, 57, and 58 are not divisible by their digits. Thus, 56 is a member of this sequence.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A038772, A244356, A237766.

SequencePosition[Table[If[NoneTrue[n/Select[IntegerDigits[n],#>0&],IntegerQ], 1,0],{n,5100}],{1,1,1}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 15 2018 *)

def a(n):
  for i in range(10**4):
    tot = 0
    for k in range(i,i+n):
      c = 0
      for b in str(k):
        if b != '0':
          if k%int(b)!=0:
            c += 1
      if c == len(str(k))-str(k).count('0'):
        tot += 1
    if tot == n:
      print(i,end=', ')
a(3)

A244358 Numbers k such that k, k+1, k+2, and k+3 are not divisible by any of their nonzero digits.

Original entry on oeis.org

56, 506, 556, 586, 596, 656, 667, 757, 787, 857, 866, 867, 956, 976, 977, 5056, 5066, 5096, 5506, 5666, 5756, 5776, 5876, 5906, 5986, 5996, 6056, 6067, 6506, 6697, 6986, 7057, 7556, 7576, 7597, 7757, 7786, 7787, 7876, 7897, 7906, 7976, 7996, 8066, 8067, 8506, 8596, 8666, 8697

Offset: 1

Derek Orr, Jun 26 2014

This is a subsequence of A244357.

All numbers end in a 6 or 7.

			56, 57, 58, and 59 are not divisible by any of their digits. Thus, 56 is a member of this sequence.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A038772, A237766, A244357.

SequencePosition[Table[If[NoneTrue[n/(IntegerDigits[n]/.(0->Nothing)),IntegerQ],1,0],{n,9000}],{1,1,1,1}][[;;,1]] (* Harvey P. Dale, Jun 06 2025 *)

def a(n):
  for i in range(10**4):
    tot = 0
    for k in range(i,i+n):
      c = 0
      for b in str(k):
        if b != '0':
          if k%int(b)!=0:
            c += 1
      if c == len(str(k))-str(k).count('0'):
        tot += 1
    if tot == n:
      print(i,end=', ')
a(4)

A382237 Numbers that are not divisible by the sum of any subset of their digits.

Original entry on oeis.org

23, 29, 34, 37, 38, 43, 46, 47, 49, 53, 56, 57, 58, 59, 67, 68, 69, 73, 74, 76, 78, 79, 83, 86, 87, 89, 94, 97, 98, 203, 223, 227, 229, 233, 239, 249, 253, 257, 263, 267, 269, 277, 283, 293, 299, 307, 323, 329, 334, 337, 338, 346, 347, 349, 353, 356, 358, 359, 367, 373, 376, 377, 379, 380, 383, 386, 388, 389, 394, 397, 398, 403

Offset: 1

Sergio Pimentel, Mar 19 2025

This sequence has density zero since no numbers with the digit '1' are in it. The sequence is infinite. Example: Numbers like 23, 203, 2003, 20003, etc. are included because none of them is divisible by 2, 3, or 5.

Conjecture: after a sufficiently large n this sequence grows faster than the prime numbers.

			358 is in the sequence because it can't be divided by 3, 5, 8, (3+5)=8, (3+8)=11, (5+8)=13 or (3+5+8)=16.
289 is not in the sequence because it can be divided by (8+9)=17.

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

Cf. A082943, A038772, A005349, A065877, A228017, A382239.

filter:= proc(n) local L,S;
  L:= convert(n,base,10);
  andmap(s -> s=0 or n mod s <> 0, map(convert,combinat:-choose(L),`+`))
end proc:
select(filter, [$1..1000]); # Robert Israel, Mar 19 2025

isok(k) = my(d=digits(k)); forsubset(#d, s, my(ss=sum(i=1, #s, d[s[i]])); if (ss && !(k % sum(i=1, #s, d[s[i]])), return(0))); return(1); \\ Michel Marcus, Mar 27 2025

from itertools import chain, combinations
def powerset(s): # skipping empty set
    return chain.from_iterable(combinations(s, r) for r in range(1, len(s)+1))
def ok(n): return all(n%t!=0 for s in powerset(list(map(int, str(n)))) if (t:=sum(s))>0)
print([k for k in range(1, 404) if ok(k)]) # Michael S. Branicky, Apr 01 2025

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

cp's OEIS Frontend

A082943 Positive numbers not divisible by any of their digits nor by the sum of their digits.

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A352380 Numbers k such that no nonzero digit of 3*k divides 3*k.

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A352381 Numbers k such that no nonzero digit of 4*k divides 4*k.

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A352382 Numbers k such that no nonzero digit of 5*k divides 5*k.

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A382239 Numbers not divisible by any of their digits nor by the sum of their digits. Digit 0 is allowed (and does not divide anything).

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A244356 Numbers n such that n and n+1 are not divisible by any of their nonzero digits.

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A244357 Numbers n such that n, n+1, and n+2 are not divisible by any of their nonzero digits.

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A244358 Numbers k such that k, k+1, k+2, and k+3 are not divisible by any of their nonzero digits.

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A352380 Numbers k such that no nonzero digit of 3k divides 3k.

A352381 Numbers k such that no nonzero digit of 4k divides 4k.

A352382 Numbers k such that no nonzero digit of 5k divides 5k.