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A273227 Consider all ways of writing the n-th composite number as the product of two divisors d1*d2 = d3*d4 = ...; a(n) is the minimum of the sums {d1 + d2, d3 + d4, ...}.

%I A273227 #18 Jul 09 2018 23:36:09
%S A273227 4,5,6,6,7,7,9,8,8,9,9,10,13,10,10,15,12,11,11,12,14,19,12,12,21,16,
%T A273227 13,13,15,14,25,14,14,15,20,17,15,16,15,22,31,16,33,16,16,18,17,21,26,
%U A273227 17,17,39,20,23,18,19,18,18,43,19,22,45,32,19,19,20,27,34
%N A273227 Consider all ways of writing the n-th composite number as the product of two divisors d1*d2 = d3*d4 = ...; a(n) is the minimum of the sums {d1 + d2, d3 + d4, ...}.
%C A273227 a(n) = A046343(n) if n is semiprime.
%C A273227 This sequence is included in A063655. - _Giovanni Resta_, May 18 2016
%C A273227 a(n) >= 2 * sqrt(A002808(n)). - _David A. Corneth_, May 20 2016
%H A273227 Charles R Greathouse IV, <a href="/A273227/b273227.txt">Table of n, a(n) for n = 1..10000</a>
%e A273227 a(14) = 10 because A002808(14) = 24 = 2*12 = 3*8 = 4*6 and 4+6 = 10 is the minimum sum.
%p A273227 with(numtheory):nn:=100:lst:={}:
%p A273227 for n from 1 to nn do:
%p A273227 it:=0:lst:={}:
%p A273227 d:=divisors(n):n0:=nops(d):
%p A273227   if n0>2 then
%p A273227   for i from 2 to n0-1 do:
%p A273227    p:=d[i]:
%p A273227     for j from i to n0-1 do:
%p A273227       q:=d[j]:
%p A273227        if p*q=n then
%p A273227         lst:=lst union {p+q}:
%p A273227         else
%p A273227        fi:
%p A273227      od:
%p A273227     od:
%p A273227     printf(`%d, `,lst[1]):
%p A273227    fi:
%p A273227    od:
%t A273227 Function[n, If[OddQ@ Length@ #, 2 Sqrt@ n, Total@ Take[#, {Length[#]/2, Length[#]/2 + 1}]] &@ Divisors@ n] /@ Select[Range@ 93, CompositeQ] (* _Michael De Vlieger_, May 20 2016 *)
%t A273227 msd[n_]:=Module[{d=Divisors[n],len},len=Length[d];If[OddQ[len], 2*d[[ (len+1)/2]], d[[len/2]]+d[[len/2+1]]]]; msd/@Select[Range[200], CompositeQ] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Jul 09 2018 *)
%o A273227 (PARI) forcomposite(n=4,100, d=divisors(n); t=#d; k=if(t%2,2*d[t\2+1], d[t\2]+d[t\2+1]); print1(k", ")) \\ _Charles R Greathouse IV_, Jun 08 2016
%Y A273227 Cf. A002808, A046343, A063655.
%K A273227 nonn
%O A273227 1,1
%A A273227 _Michel Lagneau_, May 18 2016
%E A273227 Name edited by _Jon E. Schoenfield_, Sep 12 2017