A117119 Number of partitions of 2*n into two odd prime powers.

1, 1, 2, 2, 3, 3, 4, 3, 4, 4, 4, 4, 5, 5, 6, 6, 6, 6, 6, 5, 6, 6, 6, 7, 8, 6, 9, 7, 6, 8, 7, 6, 8, 7, 7, 9, 8, 7, 9, 8, 7, 11, 9, 7, 12, 8, 7, 9, 9, 8, 10, 8, 9, 12, 11, 9, 12, 9, 8, 13, 9, 8, 13, 10, 11, 14, 11, 8, 13, 12, 10, 13, 9, 9, 16, 10, 11, 14, 10, 10, 15, 10, 9, 16, 12, 9, 16, 12, 11, 18

Offset: 1

Author

Reinhard Zumkeller, Apr 15 2006

Keywords

• nonn

Comments

Conjecture: For all n, a(n) > 0; a(n) > A002375(n).

Examples

			a(1) = #{1+1} = 1; a(2) = #{1+3} = 1; a(3) = #{1+5, 3+3} = 2;
a(20) = #{3+37, 3^2+31, 11+29, 13+3^3, 17+23} = 5;
a(21) = #{1+41, 5+37, 11+31, 13+29, 17+5^2, 19+23} = 6.
		

Links

• Eric Weisstein's World of Mathematics, Goldbach Conjecture
• Wikipedia, Goldbach's conjecture
• Wikipedia, Waring-Goldbach problem
• Index entries for sequences related to Goldbach conjecture

Crossrefs

Cf. A061345.

Programs

Maple

isA061345 := proc(n)
    if n = 1 then
        true;
    elif type(n,'even') then
        false;
    elif nops(numtheory[factorset](n)) = 1 then
        true;
    else
        false;
    end if;
end proc:
A117119 := proc(n)
    local a,j,i;
    a := 0 ;
    for i from 1 do
        j := 2*n-i ;
        if j < i then
            break;
        end if;
        if isA061345(i) and isA061345(j) then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
seq(A117119(n),n=1..60) ; # R. J. Mathar, Jul 09 2016

Mathematica

oppQ[n_] := n == 1 || OddQ[n] && PrimeNu[n] == 1; a[n_] := (k = 0; For[i = 1, True, i++, j = 2n - i; If[j < i, Break[]]; If[oppQ[i] && oppQ[j], k++] ]; k); Array[a, 100] (* Jean-François Alcover, Feb 13 2018 *)