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A046346 Composite numbers that are divisible by the sum of their prime factors (counted with multiplicity).

%I A046346 #57 May 29 2022 21:53:09
%S A046346 4,16,27,30,60,70,72,84,105,150,180,220,231,240,256,286,288,308,378,
%T A046346 440,450,476,528,540,560,576,588,594,624,627,646,648,650,728,800,805,
%U A046346 840,884,897,900,945,960,1008,1040,1056,1080,1100,1122,1134,1160,1170,1248
%N A046346 Composite numbers that are divisible by the sum of their prime factors (counted with multiplicity).
%C A046346 If m is in the sequence and d|m, then m^d is also a term. Note that this sequence contains all infinite subsequences of the form p^(p^k) for k>0, where p is a prime. - _Amiram Eldar_ and _Thomas Ordowski_, Feb 06 2019
%C A046346 If one selects some composite k, k >= 8, and decomposes (k - sopfr(k)) into an additive partition having only prime parts, then those parts, when taken as a product with k, yield an element of this sequence. - _Christopher Hohl_, Jul 30 2019
%H A046346 François Huppé, <a href="/A046346/b046346.txt">Table of n, a(n) for n = 1..50000</a> (terms 1..1000 from T. D. Noe)
%H A046346 K. Alladi and P. Erdős, <a href="http://projecteuclid.org/euclid.pjm/1102811427">On an additive arithmetic function</a>, Pacific J. Math., Volume 71, Number 2 (1977), 275-294. See "special numbers" on page 287.
%e A046346 a(38) = 884 = 2 * 2 * 13 * 17 -> 2 + 2 + 13 + 17 = 34 so 884 / 34 = 26.
%p A046346 isA046346 := proc(n)
%p A046346     if isprime(n) then
%p A046346         false;
%p A046346     elif modp(n,A001414(n)) = 0 then
%p A046346         true;
%p A046346     else
%p A046346         false;
%p A046346     end if;
%p A046346 end proc:
%p A046346 for n from 2 to 1000 do
%p A046346     if isA046346(n) then
%p A046346         printf("%d,",n);
%p A046346     end if;
%p A046346 end do: # _R. J. Mathar_, Jan 12 2016
%t A046346 Select[Range[2,1170],!PrimeQ[#]&&IntegerQ[#/Total[Times@@@FactorInteger[#]]]&] (* _Jayanta Basu_, Jun 02 2013 *)
%o A046346 (PARI) sopfr(n) = {my(f=factor(n)); sum(k=1, #f~, f[k,1]*f[k,2]);}
%o A046346 lista(nn) = forcomposite(n=2, nn, if (! (n % sopfr(n)), print1(n, ", "));); \\ _Michel Marcus_, Jan 06 2016
%o A046346 (MATLAB) m=1;for u=2:1200 if and(isprime(u)==0,mod(u,sum(factor(u)))==0); sol(m)=u; m=m+1; end; end;sol % _Marius A. Burtea_, Jul 31 2019
%o A046346 (Magma) [k:k in [2..1200]| not IsPrime(k) and  k mod (&+[m[1]*m[2]: m in Factorization(k)]) eq 0]; // _Marius A. Burtea_, Jul 31 2019
%o A046346 (Python)
%o A046346 from sympy import factorint
%o A046346 def ok(n):
%o A046346   f = factorint(n)
%o A046346   return sum(f[p] for p in f) > 1 and n % sum(p*f[p] for p in f) == 0
%o A046346 print(list(filter(ok, range(1250)))) # _Michael S. Branicky_, Apr 16 2021
%Y A046346 Cf. A036844, A046347, A046348, A001414.
%Y A046346 Contains A071142.
%K A046346 nonn
%O A046346 1,1
%A A046346 _Patrick De Geest_, Jun 15 1998
%E A046346 Description corrected by Robert A. Stump (bee_ess107(AT)yahoo.com), Jan 09 2002