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Composites k such that the concatenation of the prime factors of k, with multiplicity, in some order is divisible by k.
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%I #26 Apr 16 2024 21:02:58

%S 24,44,52,105,114,152,176,348,378,474,548,576,612,636,1518,1908,1911,

%T 2688,3168,3204,3425,3905,4704,5292,5824,6372,7695,7824,7868,7928,

%U 8064,8208,8352,8398,9072,10128,10296,10302,12467,17424,24424,25662,25872,26712,26816,27808,28749,29484,30912,31356

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

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

%H Chai Wah Wu, <a href="/A322843/b322843.txt">Table of n, a(n) for n = 1..802</a> (terms 1..329 from Robert Israel, terms 330..500 from Michael S. Branicky)

%e 52 is in the sequence because 52 = 2*2*13 and 2132 is divisible by 52.

%e 105 is in the sequence because 105 = 3*5*7 and 735 is divisible by 105.

%p filter:= proc(n) local L, P, t;

%p if isprime(n) then return false fi;

%p L:= map(t -> t[1]$t[2],ifactors(n)[2]);

%p ormap(t -> (op(sscanf(cat(op(t)),"%d"))/n)::integer, combinat:-permute(L))

%p end proc:

%p select(filter, [$4..50000]);

%t divQ@n_:=AnyTrue[(FromDigits@Flatten@IntegerDigits@#)&/@ (Permutations@Flatten@(ConstantArray @@#&/@ FactorInteger@n)),Divisible[#, n] &];

%t Cases[Range@50000,_?(CompositeQ@#&&divQ@# &)] (* _Hans Rudolf Widmer_, Jan 15 2024 *)

%o (Python)

%o from sympy import factorint, isprime

%o from sympy.utilities.iterables import multiset_permutations as MP

%o def ok(n):

%o if n < 4 or isprime(n): return False

%o mpf = []; [mpf.extend([str(p)]*e) for p, e in factorint(n).items()]

%o return any(int("".join(p))%n == 0 for p in MP(mpf))

%o print([k for k in range(32000) if ok(k)]) # _Michael S. Branicky_, Jan 19 2024

%Y Cf. A002808.

%K nonn,base

%O 1,1

%A _J. M. Bergot_ and _Robert Israel_, Dec 28 2018