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A211219 Maximum value of sigma(x) * sigma(y) * sigma(z), where x + y + z = n.

%I A211219 #13 Dec 26 2013 08:44:12
%S A211219 1,3,9,27,36,63,84,147,196,343,336,588,576,1008,864,1728,1152,2160,
%T A211219 1872,2700,2340,4032,2925,5040,4368,6300,5460,9408,6552,11760,10192,
%U A211219 14112,10080,21952,12096,18816,18816,24304,16128,30576,20832,32928,26208,33852
%N A211219 Maximum value of sigma(x) * sigma(y) * sigma(z), where x + y + z = n.
%H A211219 Paolo P. Lava, <a href="/A211219/b211219.txt">Table of n, a(n) for n = 3..500</a>
%e A211219 For n=79 the maximum product is reached when 24, 27 and 28 (24+27+28=79) are considered: sigma(24)*sigma(27)*sigma(28) = 60*40*56 = 134400.
%e A211219 For n=83 the maximum product is reached when 24, 24 and 35 (24+24+35=83) are considered: sigma(24)*sigma(24)*sigma(35) = 60*60*48 = 172800.
%p A211219 with(numtheory);
%p A211219 A211219:=proc(i)
%p A211219 local a,b,c,d,m,n,s;
%p A211219 for n from 3 to i do
%p A211219   s:=0; a:=0; b:=0; c:=n;
%p A211219   while a<=floor(n/3) do
%p A211219     while b<=floor((n-a)/2) do
%p A211219       for m from 1 to 3 do d:=sigma(a)*sigma(b)*sigma(c); od;
%p A211219       if d>s then s:=d; fi; b:=b+1; c:=c-1;
%p A211219     od;
%p A211219     a:=a+1; b:=a; c:=n-a-b;
%p A211219   od;
%p A211219   print(s);
%p A211219 od; end:
%p A211219 A211219(1000);
%t A211219 a[n_] := Max[Times @@ DivisorSigma[1, #]& /@ IntegerPartitions[n, {3}]]; Table[a[n], {n, 3, 100}] (* _Jean-François Alcover_, Dec 26 2013 *)
%Y A211219 Cf. A211216-A211218.
%K A211219 nonn
%O A211219 3,2
%A A211219 _Paolo P. Lava_, Apr 05 2012