A117119 - OEIS

%I #20 Feb 16 2025 08:33:00

%S 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,

%T 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,

%U 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

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

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

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GoldbachConjecture.html">Goldbach Conjecture</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Goldbach%27s_conjecture">Goldbach's conjecture</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Waring%E2%80%93Goldbach_problem">Waring-Goldbach problem</a>

%H <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>

%e a(1) = #{1+1} = 1; a(2) = #{1+3} = 1; a(3) = #{1+5, 3+3} = 2;

%e a(20) = #{3+37, 3^2+31, 11+29, 13+3^3, 17+23} = 5;

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

%p isA061345 := proc(n)

%p     if n = 1 then

%p         true;

%p     elif type(n,'even') then

%p         false;

%p     elif nops(numtheory[factorset](n)) = 1 then

%p         true;

%p     else

%p         false;

%p     end if;

%p end proc:

%p A117119 := proc(n)

%p     local a,j,i;

%p     a := 0 ;

%p     for i from 1 do

%p         j := 2*n-i ;

%p         if j < i then

%p             break;

%p         end if;

%p         if isA061345(i) and isA061345(j) then

%p             a := a+1 ;

%p         end if;

%p     end do:

%p     a ;

%p end proc:

%p seq(A117119(n),n=1..60) ; # _R. J. Mathar_, Jul 09 2016

%t 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 *)

%Y Cf. A061345.

%K nonn,changed

%O 1,3

%A _Reinhard Zumkeller_, Apr 15 2006