A335187 Total area of all trapezoids with bases p+q, q-p and height p*q with p,q prime, n = p+q and p < q.

0, 0, 0, 0, 18, 0, 50, 75, 98, 147, 0, 245, 242, 363, 338, 1112, 0, 1692, 578, 2050, 722, 2528, 0, 5687, 1058, 4114, 0, 5824, 0, 11431, 1682, 7216, 1922, 12907, 0, 23706, 0, 6727, 2738, 22351, 0, 38400, 3362, 27119, 3698, 28249, 0, 68387, 4418, 55626, 0, 48879, 0, 103031, 5618

Offset

1,5

Comments

Consider the Goldbach partitions of n into two distinct parts (p,q) with p < q. Then a(n) is the sum of all the products, p*q^2, using the corresponding parts from each (p,q) pair.

Total area of all rectangles with dimensions p*q X q such that n = p+q, p and q prime and p < q.

Links

Table of n, a(n) for n=1..55.

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..floor((n-1)/2)} i * (n-i)^2 * c(i) * c(n-i), where c is the prime characteristic (A010051).

Example

a(16) = 1112; There are two Goldbach partitions of 16: (3,13) and (5,11). So 3*13^2 + 5*11^2 = 507 + 605 = 1112.
a(17) = 0; There are no Goldbach partitions of 17.
a(18) = 1692; There are two Goldbach partitions of 18: (5,13) and (7,11). So 5*13^2 + 7*11^2 = 845 + 847 = 1692.
a(19) = 578; There is one Goldbach partition of 19: (2,17). Then 2*17^2 = 578.

Mathematica

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

Crossrefs

Cf. A010051.

Sequence in context: A243911 A289660 A096306 * A231962 A052441 A221394

Adjacent sequences: A335184 A335185 A335186 * A335188 A335189 A335190

Keyword

nonn,easy

Author

Wesley Ivan Hurt, May 25 2020

Status

approved