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%I #7 May 25 2020 23:48:06
%S 0,0,0,0,18,0,50,75,98,147,0,245,242,363,338,1112,0,1692,578,2050,722,
%T 2528,0,5687,1058,4114,0,5824,0,11431,1682,7216,1922,12907,0,23706,0,
%U 6727,2738,22351,0,38400,3362,27119,3698,28249,0,68387,4418,55626,0,48879,0,103031,5618
%N Total area of all trapezoids with bases p+q, q-p and height p*q with p,q prime, n = p+q and p < q.
%C Consider the Goldbach partitions of n into two distinct parts (p,q) with p < q. Then a(n) is the sum of all the products, p*q^2, using the corresponding parts from each (p,q) pair.
%C Total area of all rectangles with dimensions p*q X q such that n = p+q, p and q prime and p < q.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GoldbachPartition.html">Goldbach Partition</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Goldbach%27s_conjecture">Goldbach's conjecture</a>
%H <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F a(n) = Sum_{i=1..floor((n-1)/2)} i * (n-i)^2 * c(i) * c(n-i), where c is the prime characteristic (A010051).
%e a(16) = 1112; There are two Goldbach partitions of 16: (3,13) and (5,11). So 3*13^2 + 5*11^2 = 507 + 605 = 1112.
%e a(17) = 0; There are no Goldbach partitions of 17.
%e a(18) = 1692; There are two Goldbach partitions of 18: (5,13) and (7,11). So 5*13^2 + 7*11^2 = 845 + 847 = 1692.
%e a(19) = 578; There is one Goldbach partition of 19: (2,17). Then 2*17^2 = 578.
%t Table[Sum[i*(n - i)^2*(PrimePi[i] - PrimePi[i - 1])*(PrimePi[n - i] - PrimePi[n - i - 1]), {i, Floor[(n - 1)/2]}], {n, 80}]
%Y Cf. A010051.
%K nonn,easy
%O 1,5
%A _Wesley Ivan Hurt_, May 25 2020
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