A334152 Number of Goldbach partitions (p,q) of 2n, such that p-2 and q-2 are both prime or both not prime.

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

Offset

1,7

Links

Table of n, a(n) for n=1..89.

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} [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(6) = 1; 2*6 = 12 has one Goldbach partition, (7,5), such that 7-2 = 5 and 5-2 = 3 (both prime).
a(7) = 2; 2*7 = 14 has two Goldbach partitions, (11,3) and (7,7), such that 11-2 = 9 and 3-2 = 1 (both not prime) as well as 7-2 = 5 (both prime).

Mathematica

Table[Sum[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, A334153.

Sequence in context: A228247 A025847 A367771 * A092130 A029298 A262520

Adjacent sequences: A334149 A334150 A334151 * A334153 A334154 A334155

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Apr 16 2020

Status

approved