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

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, 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

Offset: 1

Author

Juri-Stepan Gerasimov, Nov 20 2010

Keywords

• nonn

Examples

			a(10)=2 because 10 = 1(refactorable) + 9(refactorable) = 2(refactorable) + 8(refactorable).
		

Links

• R. J. Mathar, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A033950.

Programs

Maple

with(numtheory);
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));
# alternative
isA033950 := proc(n)
    if modp(n,numtheory[tau](n)) = 0 then
        true;
    else
        false;
    end if;
end proc:
A172398 := proc(n)
    local a;
    a := 0 ;
    for i from 1 to n/2 do
        if isA033950(i) and isA033950(n-i) then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Jul 21 2015

Mathematica

a[n_] := IntegerPartitions[n, {2}, Select[Range[n], Divisible[#, DivisorSigma[0, #]]&]] // Length;
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jun 04 2023 *)

Formula

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

Extensions

Corrected by D. S. McNeil, Nov 20 2010