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A324210 Squarefree numbers k such that the sum of the distinct prime factors of k is twice the difference between the largest and the smallest prime factors of k.

%I A324210 #23 May 28 2019 19:36:43
%S A324210 110,182,374,494,782,1334,2294,3182,3854,4982,6254,7905,7917,8174,
%T A324210 9782,11534,12765,14774,15810,15834,18705,19982,20757,21614,22330,
%U A324210 22454,24182,25530,27265,28210,30381,30597,32637,35894,37410,40205,41181,41514,43005,47414,49210
%N A324210 Squarefree numbers k such that the sum of the distinct prime factors of k is twice the difference between the largest and the smallest prime factors of k.
%C A324210 This sequence is a primitive subsequence of A200070. If p|a(n) for some prime p then p*a(n) is in A200070.
%C A324210 From _Robert Israel_, Apr 09 2019: (Start)
%C A324210 All terms have at least three prime factors.
%C A324210 The number of prime factors is odd if and only if the term is even.
%C A324210 The terms with three prime factors are 2*A111192. (End)
%H A324210 Robert Israel, <a href="/A324210/b324210.txt">Table of n, a(n) for n = 1..10000</a>
%e A324210 110 = 2 * 5 * 11 is squarefree. The minimal and maximal prime divisors of 110 are 2 and 11 respectively. Twice their difference is 2 * (11-2) = 18 which is also the sum of the distinct prime divisors of 110; 2 + 5 + 11 = 18.
%p A324210 filter:= proc(n) local P;
%p A324210 if not numtheory:-issqrfree(n) then return false fi;
%p A324210 P:= numtheory:-factorset(n);
%p A324210   convert(P, `+`) = 2*(max(P)-min(P))
%p A324210 end proc:
%p A324210 select(filter, [$1..50000]);# _Robert Israel_, Apr 09 2019
%t A324210 Select[Select[Range[2, 5*10^4], SquareFreeQ], Total@ # == 2 (Last@ # - First@ #) &@ FactorInteger[#][[All, 1]] &] (* _Michael De Vlieger_, Apr 11 2019 *)
%o A324210 (PARI) is(n) = if(!issquarefree(n), return(0)); my(f=factor(n)[, 1]~); sum(i=1, #f, f[i])==2*(f[#f]-f[1])
%o A324210 forcomposite(c=1, 50000, if(is(c), print1(c, ", "))) \\ _Felix Fröhlich_, Apr 11 2019
%Y A324210 Cf. A200070, A111192.
%K A324210 nonn
%O A324210 1,1
%A A324210 _David A. Corneth_, Apr 09 2019