A342301 Number of Goldbach partitions of 2n into an odd-indexed prime and an even-indexed prime.

0, 0, 0, 1, 0, 1, 1, 0, 2, 1, 0, 3, 1, 0, 3, 0, 2, 2, 1, 1, 2, 2, 1, 3, 2, 1, 4, 0, 1, 5, 1, 1, 4, 1, 3, 2, 2, 4, 3, 2, 1, 5, 3, 2, 3, 2, 3, 4, 0, 3, 6, 2, 3, 1, 5, 7, 0, 4, 4, 5, 2, 3, 4, 2, 5, 2, 3, 4, 4, 4, 2, 5, 4, 4, 3, 3, 4, 7, 1, 3, 7, 3, 2, 5, 5, 3, 7, 2, 3, 8, 3, 2, 9

Offset

1,9

Links

Table of n, a(n) for n=1..93.

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

Example

a(12) = 3; 2*12 = 24 has 3 Goldbach partitions with one even-indexed prime and one odd-indexed prime: (19,5), (17,7) and (13,11). For example, 19 is the 8th prime and 5 is the 3rd prime, 17 is the 7th prime and 7 is the 4th prime, and 13 is the 6th prime while 11 is the 5th prime.

Mathematica

Table[Sum[Sign[Mod[PrimePi[k] + 1, 2] Mod[PrimePi[2 n - k], 2] + Mod[PrimePi[k], 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. A344981, A344982.

Sequence in context: A113214 A280494 A168016 * A029323 A194849 A071802

Adjacent sequences: A342298 A342299 A342300 * A342302 A342303 A342304

Keyword

nonn

Author

Wesley Ivan Hurt, Jun 04 2021

Status

approved