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Total area of all triangles such that p + q = 2*n, p < q (p, q prime), with base (q - p) and height p.
1

%I #9 Apr 27 2020 12:31:26

%S 0,0,0,3,6,5,12,30,34,42,54,81,72,78,126,78,168,224,84,201,332,279,

%T 252,571,462,339,760,441,522,1002,312,915,1194,282,1116,1410,810,1233,

%U 1778,1092,1206,2332,1218,1212,3396,1536,1404,3017,1572,2250,3988,2055

%N Total area of all triangles such that p + q = 2*n, p < q (p, q prime), with base (q - p) and height p.

%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} i * (n-i) * c(i) * c(2*n-i), where c is the prime characteristic (A010051).

%e a(4) = 3; 2*4 = 8 has one Goldbach partition into distinct parts (5,3). The area of the triangle is then (5 - 3)*3/2 = 3.

%e a(8) = 30; 2*8 = 16 has two Goldbach partitions into distinct parts, (13,3) and (11,5). The sum of the areas of the two triangles is (13 - 3)*3/2 + (11 - 5)*5/2 = 15 + 15 = 30.

%t Table[Sum[i (n - i) (PrimePi[i] - PrimePi[i - 1]) (PrimePi[2 n - i] - PrimePi[2 n - i - 1]), {i, n - 1}], {n, 60}]

%Y Cf. A010051.

%K nonn,easy

%O 1,4

%A _Wesley Ivan Hurt_, Apr 13 2020