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A218852 Numbers n for which sigma(n) = sigma(x) + sigma(y) + sigma(z), where n = x + y + z, with x, y, z all positive.

%I A218852 #21 Jun 22 2020 17:00:43
%S A218852 5,7,10,13,14,15,16,19,20,21,25,26,27,28,31,32,33,34,35,38,39,40,42,
%T A218852 43,44,45,46,49,50,51,52,54,55,56,57,58,61,62,63,64,65,66,68,69,70,73,
%U A218852 74,75,76,77,78,80,81,82,84,85,86,87,88,90,91,92,93,94,95,96
%N A218852 Numbers n for which sigma(n) = sigma(x) + sigma(y) + sigma(z), where n = x + y + z, with x, y, z all positive.
%C A218852 Contains the greater of every twin prime pair.
%H A218852 Harvey P. Dale, <a href="/A218852/b218852.txt">Table of n, a(n) for n = 1..400</a>
%e A218852 sigma(1) + sigma(1) + sigma(3) = sigma(5) = 6.
%e A218852 sigma(2) + sigma(2) + sigma(6) = sigma(10) = 18.
%e A218852 *sigma(2) + sigma(8) + sigma(30) = sigma(40) = 90.
%e A218852 *sigma(6) + sigma(10) + sigma(24) = sigma(40) = 90.
%e A218852 sigma(8) + sigma(8) + sigma(24) = sigma(40) = 90.
%e A218852 Hence, 5, 10 and 40 are in the sequence.
%e A218852 Note that (*) means that (x+y+z) divides xyz as well.
%p A218852 isA218852 := proc(n)
%p A218852     local x,y,z ;
%p A218852     for x from 1 to n-2 do
%p A218852         for y from x to n-x-1 do
%p A218852             z := n-x-y ;
%p A218852             if numtheory[sigma](x)+numtheory[sigma](y)+numtheory[sigma](z) = numtheory[sigma](n) then
%p A218852                 return true;
%p A218852             end if;
%p A218852         end do:
%p A218852     end do:
%p A218852     return false;
%p A218852 end proc:
%p A218852 for n from 3 to 120 do
%p A218852     if isA218852(n) then
%p A218852         printf("%d,",n);
%p A218852     end if;
%p A218852 end do: # _R. J. Mathar_, Nov 07 2012
%t A218852 xyzQ[n_]:=Module[{ips=Total/@(DivisorSigma[1,#]&/@IntegerPartitions[n,{3}])},Total[Boole[DivisorSigma[1,n]==#&/@ips]]>0]; Select[Range[ 100], xyzQ] (* _Harvey P. Dale_, Jun 22 2020 *)
%Y A218852 Cf. A000203, A211223, A218980.
%K A218852 nonn
%O A218852 1,1
%A A218852 _Jon Perry_, Nov 07 2012