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

1,10

Links

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

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

Example

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

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 *)

Crossrefs

Cf. A033950.

Sequence in context: A099200 A358351 A093578 * A070107 A299908 A044933

Adjacent sequences: A172395 A172396 A172397 * A172399 A172400 A172401

Keyword

nonn

Author

Juri-Stepan Gerasimov, Nov 20 2010

Extensions

Corrected by D. S. McNeil, Nov 20 2010

Status

approved