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A228553 Sum of the products formed by multiplying together the smaller and larger parts of each Goldbach partition of 2n.

%I A228553 #31 Feb 02 2023 15:28:24
%S A228553 0,4,9,15,46,35,82,94,142,142,263,357,371,302,591,334,780,980,578,821,
%T A228553 1340,785,1356,1987,1512,1353,2677,1421,2320,4242,1955,2803,4362,1574,
%U A228553 4021,5298,4177,4159,6731,4132,5593,9808
%N A228553 Sum of the products formed by multiplying together the smaller and larger parts of each Goldbach partition of 2n.
%C A228553 Since the product of each prime pair is semiprime and since we are adding A045917(n) of these, a(n) is expressible as the sum of exactly A045917(n) distinct semiprimes.
%H A228553 Robert Israel, <a href="/A228553/b228553.txt">Table of n, a(n) for n = 1..10000</a>
%H A228553 <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%H A228553 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A228553 a(n) = Sum_{i=2..n} c(i) * c(2*n-i) * i * (2*n-i), where c = A010051.
%F A228553 a(n) = Sum_{k=(n^2-n+2)/2..(n^2+n-2)/2} c(A105020(k)) * A105020(k), where c = A064911. - _Wesley Ivan Hurt_, Sep 19 2021
%e A228553 a(5) = 46. 2*5 = 10 has two Goldbach partitions: (7,3) and (5,5). Taking the products of the larger and smaller parts of these partitions and adding, we get 7*3 + 5*5 = 46.
%p A228553 with(numtheory); seq(sum( (2*k*i-i^2) * (pi(i)-pi(i-1)) * (pi(2*k-i)-pi(2*k-i-1)),  i=2..k), k=1..70);
%p A228553 # Alternative:
%p A228553 f:= proc(n)
%p A228553   local S;
%p A228553   S:= select(t -> isprime(t) and isprime(2*n-t), [seq(i,i=3..n,2)]);
%p A228553   add(t*(2*n-t),t=S)
%p A228553 end proc:
%p A228553 f(2):= 4:
%p A228553 map(f, [$1..200]); # _Robert Israel_, Nov 29 2020
%t A228553 c[n_] := Boole[PrimeQ[n]];
%t A228553 a[n_] := Sum[c[i]*c[2n-i]*i*(2n-i), {i, 2, n}];
%t A228553 Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Feb 02 2023 *)
%Y A228553 Cf. A010051, A045917, A064911, A105020, A185297, A187129.
%K A228553 nonn
%O A228553 1,2
%A A228553 _Wesley Ivan Hurt_, Aug 25 2013