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, 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

Links

Table of n, a(n) for n=1..49.

Eric Weisstein's World of Mathematics, Goldbach Partition

Wikipedia, Goldbach's conjecture

Index entries for sequences related to Goldbach conjecture

Index entries for sequences related to partitions

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