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 A334119 Total area of all triangles such that p + q = 2*n, p < q (p, q prime), with base (q - p) and height q. 0
 0, 0, 0, 5, 14, 7, 44, 98, 74, 158, 254, 231, 344, 258, 294, 434, 920, 856, 372, 959, 1180, 1613, 1772, 2357, 2438, 1689, 2696, 2303, 2610, 3318, 2168, 5549, 5538, 1758, 5324, 6366, 6146, 7355, 9610, 5628, 6830, 10940, 9962, 6180, 13524, 9320, 8748, 13015, 4308 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 LINKS Eric Weisstein's World of Mathematics, Goldbach Partition Wikipedia, Goldbach's conjecture FORMULA 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). EXAMPLE a(4) = 5; 2*4 = 8 has one Goldbach partition: (5,3). The area of the triangle is (5 - 3)*5/2 = 5. 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. MATHEMATICA 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}] CROSSREFS Cf. A010051, A334079. Sequence in context: A051542 A083660 A003079 * A205128 A292249 A205134 Adjacent sequences:  A334116 A334117 A334118 * A334120 A334121 A334122 KEYWORD nonn,easy AUTHOR Wesley Ivan Hurt, Apr 14 2020 STATUS approved

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Last modified November 30 02:42 EST 2020. Contains 338781 sequences. (Running on oeis4.)