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A273777 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, ...}.

%I A273777 #21 Sep 13 2017 02:15:22
%S A273777 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 A273777 16,22,23,24,18,25,26,14,27,20,28,29,16,30,22,31,32,33,24,34,18,35,36,
%U A273777 26,37,38,39,28,40,18,41,42,30,43,44,22,45,32,46,47,20,48
%N A273777 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 A273777 The divisors must be > 1 and < n.
%C A273777 For the minimum sums see A273227.
%F A273777 Let m = A002808(n). Then a(n) = A020639(m) + m / A020639(m).
%e A273777 a(14) = 14 because A002808(14) = 24 = 2*12 = 3*8 = 4*6 and 2+12 = 14 is the maximum sum.
%p A273777 with(numtheory):nn:=100:lst:={}:
%p A273777 for n from 1 to nn do:
%p A273777 it:=0:lst:={}:
%p A273777 d:=divisors(n):n0:=nops(d):
%p A273777   if n0>2 then
%p A273777   for i from 2 to n0-1 do:
%p A273777    p:=d[i]:
%p A273777     for j from i to n0-1 do:
%p A273777       q:=d[j]:
%p A273777        if p*q=n then
%p A273777         lst:=lst union {p+q}:
%p A273777         else
%p A273777        fi:
%p A273777      od:
%p A273777     od:
%p A273777     n0:=nops(lst):printf(`%d, `, lst[n0]):
%p A273777    fi:
%p A273777    od:
%t A273777 Function[n, Max@ Map[Plus[#, n/#] &, Rest@ Take[#, Ceiling[Length[#]/2]]] &@ Divisors@ n] /@ Select[Range@ 120, CompositeQ] (* _Michael De Vlieger_, May 30 2016 *)
%o A273777 (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 A273777 Cf. A002808, A020639, A046343, A063655, A273227.
%K A273777 nonn
%O A273777 1,1
%A A273777 _Michel Lagneau_, May 30 2016
%E A273777 Name edited by _Jon E. Schoenfield_, Sep 12 2017