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%I #12 May 04 2023 15:56:14
%S 0,0,1,0,0,0,0,1,1,1,0,0,0,0,1,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,2,1,0,
%T 0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,1,0,0,0,2,1,2,0,0,0,0,1,2,0,0,0,0,0,
%U 2,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,2
%N Number of ways to represent sigma(n) as sigma(x) + sigma(y) where x+y = n.
%C From an idea of Charles R Greathouse IV.
%C a(A211223(n)) > 0. - _Reinhard Zumkeller_, Jan 06 2013
%H Paolo P. Lava, <a href="/A211225/b211225.txt">Table of n, a(n) for n = 1..10000</a>
%e a(3)=1 because sigma(3)=sigma(1)+sigma(2)=4;
%e a(32)=2 because sigma(32)=sigma(4)+sigma(28)=sigma(14)+sigma(18)=63;
%e a(117)=3 because sigma(117)=sigma(41)+sigma(76)=sigma(52)+sigma(65)=sigma(56)+sigma(61)=182; etc.
%p with(numtheory);
%p A211225:=proc(q)
%p local b,i,n;
%p for n from 1 to q do
%p b:=0;
%p for i from 1 to trunc(n/2) do
%p if sigma(i)+sigma(n-i)=sigma(n) then b:=b+1; fi;
%p od;
%p print(b)
%p od; end:
%p A211225(1000);
%t a[n_] := With[{s = DivisorSigma[1, n]}, Sum[Boole[s == DivisorSigma[1, x] + DivisorSigma[1, n-x]], {x, 1, Quotient[n, 2]}]];
%t Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, May 04 2023 *)
%o (PARI) a(n)=my(t=sigma(n)); sum(i=1, n\2, sigma(i)+sigma(n-i)==t) \\ _Charles R Greathouse IV_, May 07 2012
%o (Haskell)
%o a211225 n = length $ filter (== a000203 n) $ zipWith (+) us' vs where
%o (us,vs@(v:_)) = splitAt (fromInteger $ (n - 1) `div` 2) a000203_list
%o us' = if even n then v : reverse us else reverse us
%o -- _Reinhard Zumkeller_, Jan 06 2013
%Y Cf. A083207, A204830, A204831, A211223, A211224.
%Y Cf. A000203.
%K nonn
%O 1,32
%A _Paolo P. Lava_, May 07 2012
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