A344982 Number of Goldbach partitions of 2n into two odd-indexed primes.

0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 2, 1, 0, 1, 1, 0, 2, 1, 0, 2, 1, 0, 2, 0, 1, 3, 0, 0, 2, 2, 0, 1, 3, 0, 2, 2, 0, 2, 3, 0, 2, 1, 1, 2, 1, 1, 2, 3, 0, 0, 6, 0, 1, 4, 0, 1, 3, 1, 1, 3, 2, 0, 2, 2, 2, 3, 1, 1, 4, 0, 2, 3, 1, 2, 2, 1, 2, 4, 2, 2, 2, 2, 2, 3, 1, 3, 2

Offset

1,11

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

Example

a(11) = 2; There are 2 Goldbach partitions of 2*11 = 22 into two odd-indexed primes: (17,5) and (11,11) where 5, 11, and 17 are primes with odd indices (i.e., 5 is the 3rd prime, 11 is the 5th prime and 17 is the 7th prime).

Mathematica

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

Crossrefs

Cf. A000720 (pi), A010051, A344981 (even-indexed).

Sequence in context: A359288 A096142 A216921 * A359240 A280285 A033719

Adjacent sequences: A344979 A344980 A344981 * A344983 A344984 A344985

Keyword

nonn

Author

Wesley Ivan Hurt, Jun 03 2021

Status

approved