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A309310 Intersection of A046346 and (A046346-2).

%I A309310 #25 Jun 11 2023 14:21:20
%S A309310 70,286,646,648,18048,26752,39128,40000,55648,60760,64798,72928,73726,
%T A309310 164736,167440,174018,298298,324478,332748,352798,361788,373246,
%U A309310 434928,649798,719998,862750,871198
%N A309310 Intersection of A046346 and (A046346-2).
%C A309310 Numbers k such that both k and k+2 are composite and each is divisible by the sum of its prime factors (counted with multiplicity).
%C A309310 There are at least two cases where k, k+2 and k+4 are all in A046346: k=646 and k=38104990.  Are there more?
%C A309310 Up to 3*10^12 there is only one other such triple for k=590269019100. - _Giovanni Resta_, Jul 24 2019
%H A309310 Amiram Eldar, <a href="/A309310/b309310.txt">Table of n, a(n) for n = 1..2000</a> (terms 1..200 from Robert Israel)
%e A309310 a(1)=70 is a term because 70=2*5*7 is divisible by 2+5+7=14 and 72=2^3*3^2 is divisible by 2*3+3*2=12.
%p A309310 filter:= proc(n) local E,t;
%p A309310   if isprime(n) then return false fi;
%p A309310   E:= ifactors(n)[2];
%p A309310   n mod add(t[1]*t[2],t=E) = 0
%p A309310 end proc:
%p A309310 A046346:= select(filter, {$2..10^6}):
%p A309310 sort(convert(A046346 intersect map(`-`,A046346,2),list));
%t A309310 Select[Partition[Select[Range[2, 10^6], And[! PrimeQ[#], IntegerQ[#/Total[Times @@@ FactorInteger[#]]]] &], 2, 1], Subtract @@ # == -2 &][[All, 1]] (* _Michael De Vlieger_, Jul 22 2019 *)
%t A309310 cdsQ[n_]:=CompositeQ[n]&&Divisible[n,Total[Times@@@FactorInteger[n]]]; SequencePosition[Table[If[cdsQ[n],1,0],{n,872000}],{1,_,1}][[All,1]] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Jul 23 2019 *)
%Y A309310 Cf. A046346.
%K A309310 nonn
%O A309310 1,1
%A A309310 _J. M. Bergot_ and _Robert Israel_, Jul 22 2019