A344981 Number of Goldbach partitions of 2n into two even-indexed primes.

0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 2, 0, 0, 2, 0, 1, 1, 1, 1, 1, 1, 1, 2, 0, 0, 3, 1, 1, 1, 1, 2, 1, 0, 2, 3, 0, 1, 2, 2, 1, 2, 0, 3, 2, 0, 2, 2, 1, 1, 2, 1, 4, 1, 0, 4, 2, 1, 3, 2, 1, 3, 0, 1, 4, 1, 1, 2, 1, 4, 3, 1, 0, 5, 1, 2, 1, 3, 3, 1, 1, 2, 4, 2, 1, 2, 3, 2, 3, 2, 3, 2, 2

Offset

1,13

Links

Table of n, a(n) for n=1..94.

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_{k=1..n} ((pi(k)+1) mod 2) * ((pi(2*n-k)+1) mod 2) * c(k) * c(2*n-k), where c(n) is the prime characteristic.

Example

a(13) = 2; There are 2 Goldbach partitions of 2*13 = 26 into two even-indexed primes: (19,7) and (13,13) where 7, 13, and 19 are primes with even indices (i.e., 7 is the 4th prime, 13 is the 6th prime and 19 is the 8th prime).

Mathematica

Table[Sum[Mod[PrimePi[k] + 1, 2] Mod[PrimePi[2 n - k] + 1, 2] (PrimePi[k] - PrimePi[k - 1]) (PrimePi[2 n - k] - PrimePi[2 n - k - 1]), {k, n}], {n, 100}]

Crossrefs

Cf. A000720 (pi), A010051, A344982 (odd-indexed).

Sequence in context: A324730 A118683 A175800 * A161116 A262726 A112605

Adjacent sequences: A344978 A344979 A344980 * A344982 A344983 A344984

Keyword

nonn

Author

Wesley Ivan Hurt, Jun 03 2021

Status

approved