A279729 Sum of all the parts of the Goldbach partitions (p,q) of 2n such that all primes from p to q (inclusive) appear as a part in some Goldbach partition of p+q = 2n.

0, 0, 6, 8, 20, 12, 14, 0, 36, 0, 22, 72, 26, 0, 90, 0, 34, 72, 38, 0, 42, 0, 46, 0, 0, 52, 0, 0, 58, 300, 62, 0, 0, 68, 0, 0, 74, 0, 78, 0, 82, 252, 86, 0, 90, 0, 94, 0, 0, 100, 0, 0, 212, 0, 0, 112, 0, 0, 118, 240, 122, 0, 0, 128, 0, 0, 134, 0, 138, 0, 142, 144, 146, 0

Offset

1,3

Links

Table of n, a(n) for n=1..74.

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) = 2n * A278700(n).

a(n) = A279727(n) + A279728(n).

Maple

with(numtheory): A279729:=n->2*n*add((pi(i)-pi(i-1)) * (pi(2*n-i)-pi(2*n-i-1)) * (product(1-abs((pi(k)-pi(k-1))-(pi(2*n-k)-pi(2*n-k-1))), k=i..n)), i=3..n): seq(A279729(n), n=1..100);

Mathematica

f[n_, x_: 0] := Sum[(If[x == 0, i, 2 n - i] Boole[PrimeQ@ i] Boole[PrimeQ[2 n - i]]) Product[1 - Abs[Boole[PrimeQ@ k] - Boole[PrimeQ[2 n - k]]], {k, i, n}], {i, 3, n}]; Table[f@ n + f[n, 1], {n, 100}] (* Michael De Vlieger, Dec 18 2016 *)

Crossrefs

Cf. A278700, A279727, A279728.

Sequence in context: A349225 A199884 A028331 * A357946 A309653 A113806

Adjacent sequences: A279726 A279727 A279728 * A279730 A279731 A279732

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Dec 17 2016

Status

approved