A133606 Numbers n with all digits different, such that all of its digits divide n, but none of the nonzero digits not in n divide n.

1, 124, 175, 412, 1326, 1362, 1395, 1935, 3126, 3162, 3195, 3915, 4172, 9315, 14728, 17248, 21784, 72184, 123864, 124368, 126384, 132648, 132864, 136824, 138624, 142368, 148632, 162384, 163248, 163824, 164328, 164832, 168432, 183264, 186432

Offset

1,2

Comments

Digit 1 must appear in each term. - Michael S. Branicky, Jul 06 2021

References

Rodolfo Kurchan, Snark, December 2007

Links

Rodolfo Kurchan, Dec 27 2007, Table of n, a(n) for n = 1..116

Example

124 is divisible by 1, 2 and 4 and it is not divisible by 3, 5, 6, 7, 8 and 9.
175 is divisible by 1, 5 and 7 and it is not divisible by 2, 3, 4, 6, 8 and 9.
The final term, 864312, is divisible by 8, 6, 4, 3, 1 and 2, but not by 5, 7 or 9.

Mathematica

Select[Range@200000, (s=IntegerDigits@#; Length@s==Length@Union@s&&
Quiet@AllTrue[#/s, IntegerQ]&&NoneTrue[#/Complement[Range@9, s], IntegerQ])&] (* Giorgos Kalogeropoulos, Jul 06 2021 *)
Select[Range[200000], DigitCount[#, 10, 0]==0&&Max[DigitCount[#]]==1 && AllTrue[ #/IntegerDigits[#], IntegerQ]&&NoneTrue[#/Complement[Range[ 9], IntegerDigits[ #]], IntegerQ]&] (* Harvey P. Dale, Dec 30 2021 *)

Prog

(Python)
def ok(n):
  s = str(n); ss = set(s)
  return '0' not in ss and len(s) == len(ss) and all(n%int(d) == 0 for d in ss) and all(n%int(d) for d in set("123456789")-ss)
print(list(filter(ok, range(200000)))) # Michael S. Branicky, Jul 06 2021
(Python) # generates entire sequence
from sympy.utilities.iterables import multiset_permutations
def agen():
  for digits in range(1, 10):
    for mp in multiset_permutations("123456789", digits):
      n, mpc = int("".join(mp)), set("123456789") - set(mp)
      if all(n%int(d) == 0 for d in mp) and all(n%int(d) for d in mpc):
        yield n
print([an for an in agen()]) # Michael S. Branicky, Jul 06 2021

Crossrefs

Cf. A133598.

Sequence in context: A056085 A129010 A196582 * A270864 A038866 A043367

Adjacent sequences: A133603 A133604 A133605 * A133607 A133608 A133609

Keyword

nonn,fini,full,base

Author

Rodolfo Kurchan, Dec 27 2007

Extensions

Name clarified by Michael S. Branicky, Jul 06 2021

Status

approved