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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
OFFSET
1,4
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
Sequence in context: A083660 A375302 A003079 * A205128 A292249 A205134
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Apr 14 2020
STATUS
approved