A279794 Number of Goldbach partitions (p,q) of 2n such that |p-q| > n.

0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 3, 2, 2, 3, 2, 2, 2, 1, 3, 3, 1, 3, 3, 3, 3, 5, 3, 2, 4, 4, 2, 4, 3, 3, 4, 1, 3, 4, 2, 4, 4, 3, 4, 5, 4, 4, 6, 2, 3, 5, 2, 4, 5, 3, 3, 4, 3, 4, 5, 2, 2, 5, 2, 4, 5, 3, 4, 5, 3, 3, 8, 5, 3, 6, 4, 4, 8, 4, 4, 7, 3, 4, 6, 5, 6, 7, 5

Offset: 1

Author

Wesley Ivan Hurt, Dec 18 2016

Keywords

• nonn
• easy

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• 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. A010051, A002375, A279792.

Programs

Maple

with(numtheory): A279794:=n->add( (pi(i)-pi(i-1)) * (pi(2*n-i)-pi(2*n-i-1)) * (1-signum(floor(n/(2*(n-i))))), i=3..n-1): seq(A279794(n), n=1..100);
# Alternative:
f:= proc(n) local p;
   nops(select(t -> isprime(t) and isprime(2*n-t), [seq(p,p=3..(n-1)/2,2)]))
end proc:
map(f, [$1..100]); # Robert Israel, Feb 15 2021

Mathematica

Table[Sum[Boole[PrimeQ@ i] Boole[PrimeQ[2 n - i]] (1 - Sign@ Floor[n/(2 (n - i))]), {i, 3, n - 1}], {n, 100}] (* Michael De Vlieger, Dec 21 2016 *)

Formula

a(n) = Sum_{i=3..n-1} A010051(i) * A010051(2n-i) * (1-sign(floor(n/(2*(n-i))))).