A322843 Composites k such that the concatenation of the prime factors of k, with multiplicity, in some order is divisible by k.

24, 44, 52, 105, 114, 152, 176, 348, 378, 474, 548, 576, 612, 636, 1518, 1908, 1911, 2688, 3168, 3204, 3425, 3905, 4704, 5292, 5824, 6372, 7695, 7824, 7868, 7928, 8064, 8208, 8352, 8398, 9072, 10128, 10296, 10302, 12467, 17424, 24424, 25662, 25872, 26712, 26816, 27808, 28749, 29484, 30912, 31356

Offset

1,1

Comments

If p is a prime with d digits that divides 10^(d+1)+1, then 4*p is in the sequence.

Links

Chai Wah Wu, Table of n, a(n) for n = 1..802 (terms 1..329 from Robert Israel, terms 330..500 from Michael S. Branicky)

Example

52 is in the sequence because 52 = 2*2*13 and 2132 is divisible by 52.
105 is in the sequence because 105 = 3*5*7 and 735 is divisible by 105.

Maple

filter:= proc(n) local L, P, t;
  if isprime(n) then return false fi;
  L:= map(t -> t[1]$t[2], ifactors(n)[2]);
  ormap(t -> (op(sscanf(cat(op(t)), "%d"))/n)::integer,  combinat:-permute(L))
end proc:
select(filter, [$4..50000]);

Mathematica

divQ@n_:=AnyTrue[(FromDigits@Flatten@IntegerDigits@#)&/@ (Permutations@Flatten@(ConstantArray @@#&/@ FactorInteger@n)), Divisible[#, n] &];
Cases[Range@50000, _?(CompositeQ@#&&divQ@# &)] (* Hans Rudolf Widmer, Jan 15 2024 *)

Prog

(Python)
from sympy import factorint, isprime
from sympy.utilities.iterables import multiset_permutations as MP
def ok(n):
    if n < 4 or isprime(n): return False
    mpf = []; [mpf.extend([str(p)]*e) for p, e in factorint(n).items()]
    return any(int("".join(p))%n == 0 for p in MP(mpf))
print([k for k in range(32000) if ok(k)]) # Michael S. Branicky, Jan 19 2024

Crossrefs

Cf. A002808.

Sequence in context: A258865 A072096 A055480 * A211568 A324459 A324457

Adjacent sequences: A322840 A322841 A322842 * A322844 A322845 A322846

Keyword

nonn,base

Author

J. M. Bergot and Robert Israel, Dec 28 2018

Status

approved