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A172398 Number of partitions of n into the sum of two refactorable numbers (A033950).

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