A279536 Count the squarefree numbers appearing in each interval [p,q] where (p,q) is a Goldbach partition of 2n such that all primes from p to q (inclusive) appear as a part in some Goldbach partition of p+q = 2n, and then add the results.

0, 0, 1, 2, 5, 3, 1, 0, 9, 0, 1, 19, 1, 0, 21, 0, 1, 10, 1, 0, 4, 0, 1, 0, 0, 3, 0, 0, 1, 68, 1, 0, 0, 5, 0, 0, 1, 0, 4, 0, 1, 25, 1, 0, 3, 0, 1, 0, 0, 3, 0, 0, 8, 0, 0, 5, 0, 0, 1, 12, 1, 0, 0, 5, 0, 0, 1, 0, 4, 0, 1, 2, 1, 0, 0, 5, 0, 0, 1, 0, 14, 0, 1, 0, 0, 5, 0, 0, 1, 0

Offset: 1

Author

Wesley Ivan Hurt, Dec 14 2016

Keywords

• nonn
• easy

Comments

a(n) >= A279315(n). - Wesley Ivan Hurt, Dec 17 2016

Links

• 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

Crossrefs

Cf. A008683, A010051, A279315.

Programs

Maple

with(numtheory): A279536:=n->add( (pi(i)-pi(i-1)) * (pi(2*n-i)-pi(2*n-i-1)) * add(mobius(j)^2, j=i..2*n-i) * (product(1-abs((pi(k)-pi(k-1))-(pi(2*n-k)-pi(2*n-k-1))), k=i..n)), i=3..n): seq(A279536(n), n=1..100);

Formula

a(n) = Sum_{i=3..n} (A010051(i) * A010051(2n-i) * (Sum_{j=i..2n-i} mu(j)^2) * (Product_{k=i..n} (1-abs(A010051(k)-A010051(2n-k))))), where mu is the Möbius function (A008683).