A334153 Number of Goldbach partitions (p,q) of 2n, such that exactly one of p-2 or q-2 is prime.

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

Offset

1,8

Links

Table of n, a(n) for n=1..88.

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=1..n} (1 - [c(i-2) = c(2*n-i-2)]) * c(i) * c(2*n-i), where [] is the Iverson bracket and c is the prime characteristic (A010051).

Example

a(5) = 1; 2*5 = 10 has one Goldbach partition, (7,3), with 7-2 = 5 (prime) and 3-2 = 1 (not prime).
a(8) = 2; 2*8 = 16 has two Goldbach partitions, (13,3) and (11,5), with 13-2 = 11 (prime) and 3-2 (not prime) as well as 11-2 = 9 (not prime) and 5-2 = 3 (prime).

Mathematica

Table[Sum[(1 - KroneckerDelta[PrimePi[i - 2] - PrimePi[i - 3], PrimePi[2 n - i - 2] - PrimePi[2 n - i - 3]])*(PrimePi[i] - PrimePi[i - 1]) (PrimePi[2 n - i] - PrimePi[2 n - i - 1]), {i, n}], {n, 100}]

Crossrefs

Cf. A010051, A045917, A334152.

Sequence in context: A287524 A157237 A065676 * A281461 A146973 A003263

Adjacent sequences: A334150 A334151 A334152 * A334154 A334155 A334156

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Apr 16 2020

Status

approved