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

1,3

Comments

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

Links

Table of n, a(n) for n=1..90.

Eric Weisstein's World of Mathematics, Goldbach Conjecture

Wikipedia, Goldbach's conjecture

Wikipedia, Waring-Goldbach problem

Index entries for sequences related to Goldbach conjecture

Example

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.

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 *)

Crossrefs

Cf. A061345.

Sequence in context: A196048 A176075 A256555 * A208280 A139141 A122953

Adjacent sequences: A117116 A117117 A117118 * A117120 A117121 A117122

Keyword

nonn,changed

Author

Reinhard Zumkeller, Apr 15 2006

Status

approved