A279315 Count the primes 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, 4, 2, 1, 0, 6, 0, 1, 12, 1, 0, 12, 0, 1, 6, 1, 0, 2, 0, 1, 0, 0, 2, 0, 0, 1, 30, 1, 0, 0, 2, 0, 0, 1, 0, 2, 0, 1, 12, 1, 0, 2, 0, 1, 0, 0, 2, 0, 0, 4, 0, 0, 2, 0, 0, 1, 6, 1, 0, 0, 2, 0, 0, 1, 0, 2, 0, 1, 2, 1, 0, 0, 2, 0, 0, 1, 0, 6, 0, 1, 0, 0, 2, 0

Offset

1,4

Links

Table of n, a(n) for n=1..87.

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} (c(i) * c(2n-i) * (pi(2*n-i)-pi(i-1)) * (Product_{k=i..n} (1-abs(c(k)-c(2n-k)))), where pi is the prime counting function (A000720), and c is the prime characteristic (A010051).

From Wesley Ivan Hurt, Dec 17 2016: (Start)

a(n) = A010051(n)*A278700(n)^2+(1-A010051(n))*A278700(n)*(A278700(n)+1).

a(n) <= A279536(n). (End)

Maple

with(numtheory): A279315:=n->add( (pi(i)-pi(i-1)) * (pi(2*n-i)-pi(2*n-i-1)) * (pi(2*n-i)-pi(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(A279315(n), n=1..100);

Mathematica

f[n_] := Sum[ Boole[PrimeQ[i]] Boole[PrimeQ[ 2n -i]] (PrimePi[ 2n -i] - PrimePi[i -1]) Product[(1 - Abs[Boole[PrimeQ[k]] - Boole[PrimeQ[ 2n -k]]]), {k, i, n}], {i, 3, n}]; Array[f, 80] (* Robert G. Wilson v, Dec 15 2016 *)

Crossrefs

Cf. A000720, A010051, A278700, A279481, A279536.

Sequence in context: A273240 A201316 A105023 * A303293 A344637 A201558

Adjacent sequences: A279312 A279313 A279314 * A279316 A279317 A279318

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Dec 13 2016

Status

approved