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

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, 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, 2, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2

Offset: 1

Author

Paolo P. Lava, May 07 2012

Keywords

• nonn

Comments

From an idea of Charles R Greathouse IV.

a(A211223(n)) > 0. - Reinhard Zumkeller, Jan 06 2013

Examples

			a(3)=1 because sigma(3)=sigma(1)+sigma(2)=4;
a(32)=2 because sigma(32)=sigma(4)+sigma(28)=sigma(14)+sigma(18)=63;
a(117)=3 because sigma(117)=sigma(41)+sigma(76)=sigma(52)+sigma(65)=sigma(56)+sigma(61)=182; etc.
		

Links

• Paolo P. Lava, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A083207, A204830, A204831, A211223, A211224.

Cf. A000203.

Programs

Haskell

a211225 n = length $ filter (== a000203 n) $ zipWith (+) us' vs where
   (us,vs@(v:_)) = splitAt (fromInteger $ (n - 1) `div` 2) a000203_list
   us' = if even n then v : reverse us else reverse us
-- Reinhard Zumkeller, Jan 06 2013

Maple

with(numtheory);
A211225:=proc(q)
local b,i,n;
for n from 1 to q do
  b:=0;
  for i from 1 to trunc(n/2) do
    if sigma(i)+sigma(n-i)=sigma(n) then b:=b+1; fi;
  od;
  print(b)
od; end:
A211225(1000);

Mathematica

a[n_] := With[{s = DivisorSigma[1, n]}, Sum[Boole[s == DivisorSigma[1, x] + DivisorSigma[1, n-x]], {x, 1, Quotient[n, 2]}]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, May 04 2023 *)

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