A334065 Total area of all triangles such that p + q = 2*n, p < q (p, q prime), with base (q + p) and height (q - p).

0, 0, 0, 8, 20, 12, 56, 128, 108, 200, 308, 312, 416, 336, 420, 512, 1088, 1080, 456, 1160, 1512, 1892, 2024, 2928, 2900, 2028, 3456, 2744, 3132, 4320, 2480, 6464, 6732, 2040, 6440, 7776, 6956, 8588, 11388, 6720, 8036, 13272, 11180, 7392, 16920, 10856, 10152, 16032

Offset

1,4

Links

Table of n, a(n) for n=1..48.

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) = 2*n * Sum_{i=1..n} (n-i) * c(i) * c(2*n-i), where c is the prime characteristic (A010051).

Example

a(4) = 8; 2*4 = 8 has one Goldbach partition: (5,3). The area of the triangle is (5 + 3)*(5 - 3)/2 = 8.
a(8) = 128; 2*8 = 16 has the two Goldbach partitions: (13,3) and (11,5). The total area of the two triangles is (13 + 3)*(13 - 3)/2 + (11 + 5)*(11 - 5)/2 = 80 + 48 = 128.

Mathematica

Table[2 n*Sum[(n - i) (PrimePi[i] - PrimePi[i - 1]) (PrimePi[2 n - i] - PrimePi[2 n - i - 1]), {i, n}], {n, 80}]

Prog

(PARI) a(n) = my(s=0); forprime(p=1, n, if (isprime(2*n-p), s+=n-p)); 2*n*s; \\ Michel Marcus, Apr 14 2020

Crossrefs

Cf. A010051.

Sequence in context: A373743 A081963 A208085 * A128909 A115147 A302241

Adjacent sequences: A334062 A334063 A334064 * A334066 A334067 A334068

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Apr 13 2020

Status

approved