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Consider all ways of writing the n-th composite number as the product of two divisors d1*d2 = d3*d4 = ... where each divisor is larger than 1; a(n) is the maximum of the sums {d1 + d2, d3 + d4, ...}.
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%I #21 Sep 13 2017 02:15:22

%S 4,5,6,6,7,8,9,8,10,11,12,10,13,14,10,15,12,16,17,18,14,19,12,20,21,

%T 16,22,23,24,18,25,26,14,27,20,28,29,16,30,22,31,32,33,24,34,18,35,36,

%U 26,37,38,39,28,40,18,41,42,30,43,44,22,45,32,46,47,20,48

%N Consider all ways of writing the n-th composite number as the product of two divisors d1*d2 = d3*d4 = ... where each divisor is larger than 1; a(n) is the maximum of the sums {d1 + d2, d3 + d4, ...}.

%C The divisors must be > 1 and < n.

%C For the minimum sums see A273227.

%F Let m = A002808(n). Then a(n) = A020639(m) + m / A020639(m).

%e a(14) = 14 because A002808(14) = 24 = 2*12 = 3*8 = 4*6 and 2+12 = 14 is the maximum sum.

%p with(numtheory):nn:=100:lst:={}:

%p for n from 1 to nn do:

%p it:=0:lst:={}:

%p d:=divisors(n):n0:=nops(d):

%p if n0>2 then

%p for i from 2 to n0-1 do:

%p p:=d[i]:

%p for j from i to n0-1 do:

%p q:=d[j]:

%p if p*q=n then

%p lst:=lst union {p+q}:

%p else

%p fi:

%p od:

%p od:

%p n0:=nops(lst):printf(`%d, `, lst[n0]):

%p fi:

%p od:

%t Function[n, Max@ Map[Plus[#, n/#] &, Rest@ Take[#, Ceiling[Length[#]/2]]] &@ Divisors@ n] /@ Select[Range@ 120, CompositeQ] (* _Michael De Vlieger_, May 30 2016 *)

%o (PARI) lista(nn) = {forcomposite(n=2, nn, m = 0; fordiv(n, d, if ((d != 1) && (d != n), m = max(m, d+n/d));); print1(m, ", "););} \\ _Michel Marcus_, Sep 13 2017

%Y Cf. A002808, A020639, A046343, A063655, A273227.

%K nonn

%O 1,1

%A _Michel Lagneau_, May 30 2016

%E Name edited by _Jon E. Schoenfield_, Sep 12 2017