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

Author

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

Keywords

• nonn
• base

Comments

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

Examples

			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.
		

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)

Crossrefs

Cf. A002808.

Programs

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 *)

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