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

1,4

Comments

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

Links

Table of n, a(n) for n=1..90.

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=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).

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);

Crossrefs

Cf. A008683, A010051, A279315.

Sequence in context: A132898 A365728 A265318 * A269954 A326953 A234255

Adjacent sequences: A279533 A279534 A279535 * A279537 A279538 A279539

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Dec 14 2016

Status

approved