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%I #27 Jun 04 2023 18:56:58
%S 0,1,1,1,0,0,0,0,1,2,1,0,1,1,0,1,1,1,1,2,1,0,0,1,1,2,1,0,0,1,0,1,1,0,
%T 0,2,1,1,0,0,1,2,0,1,1,0,0,3,1,0,0,1,0,1,0,0,1,2,0,1,1,1,0,2,1,0,0
%N Number of partitions of n into the sum of two refactorable numbers (A033950).
%H R. J. Mathar, <a href="/A172398/b172398.txt">Table of n, a(n) for n = 1..1000</a>
%F a(n) = Sum_{i=1..floor(n/2)} ((1+floor(i/d(i)) - ceiling(i/d(i))) * (1 + floor((n-i)/d(n-i)) - ceiling((n-i)/d(n-i)))). - _Wesley Ivan Hurt_, Jan 12 2013
%e a(10)=2 because 10 = 1(refactorable) + 9(refactorable) = 2(refactorable) + 8(refactorable).
%p with(numtheory);
%p a:=n-> sum( ((1 + floor(i/tau(i)) - ceil(i/tau(i))) * (1 + floor((n-i)/tau(n-i)) - ceil((n-i)/tau(n-i))) ), i=1..floor(n/2));
%p # alternative
%p isA033950 := proc(n)
%p if modp(n,numtheory[tau](n)) = 0 then
%p true;
%p else
%p false;
%p end if;
%p end proc:
%p A172398 := proc(n)
%p local a;
%p a := 0 ;
%p for i from 1 to n/2 do
%p if isA033950(i) and isA033950(n-i) then
%p a := a+1 ;
%p end if;
%p end do:
%p a ;
%p end proc: # _R. J. Mathar_, Jul 21 2015
%t a[n_] := IntegerPartitions[n, {2}, Select[Range[n], Divisible[#, DivisorSigma[0, #]]&]] // Length;
%t Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Jun 04 2023 *)
%Y Cf. A033950.
%K nonn
%O 1,10
%A _Juri-Stepan Gerasimov_, Nov 20 2010
%E Corrected by _D. S. McNeil_, Nov 20 2010
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