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%I #8 Apr 27 2020 13:08:57
%S 0,0,0,5,14,7,44,98,74,158,254,231,344,258,294,434,920,856,372,959,
%T 1180,1613,1772,2357,2438,1689,2696,2303,2610,3318,2168,5549,5538,
%U 1758,5324,6366,6146,7355,9610,5628,6830,10940,9962,6180,13524,9320,8748,13015,4308
%N Total area of all triangles such that p + q = 2*n, p < q (p, q prime), with base (q - p) and height 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..n-1} (n-i) * (2*n-i) * c(i) * c(2*n-i), where c is the prime characteristic (A010051).
%e a(4) = 5; 2*4 = 8 has one Goldbach partition: (5,3). The area of the triangle is (5 - 3)*5/2 = 5.
%e a(8) = 98; 2*8 = 16 has two Goldbach partitions: (13,3) and (11,5). The sum of the areas is (13 - 3)*13/2 + (11 - 5)*11/2 = 65 + 33 = 98.
%t Table[Sum[(n - i) (2 n - i) (PrimePi[i] - PrimePi[i - 1]) (PrimePi[2 n - i] - PrimePi[2 n - i - 1]), {i, n - 1}], {n, 60}]
%Y Cf. A010051, A334079.
%K nonn,easy
%O 1,4
%A _Wesley Ivan Hurt_, Apr 14 2020
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