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A211225 Number of ways to represent sigma(n) as sigma(x) + sigma(y) where x+y = n.

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