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
Eric Weisstein's World of Mathematics, Goldbach Partition
Wikipedia, Goldbach's conjecture
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
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, May 25 2020
STATUS
approved