A302231 Number of pairs of Goldbach partitions of 2n, (p,q) and (s,t) with p < s <= t < q such that s = p + 2 and t = q - 2.

0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 2, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 2, 0, 1, 2, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 1, 1, 0, 1, 3, 0, 0, 2, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 1, 2, 0, 0, 2, 0, 1, 2, 0, 0, 2, 0, 0, 3, 0, 0

Offset

1,24

Comments

Number of tangency points on all circles with radius sqrt(2) and center (p,q) where p and q are prime, p + q = 2n and p <= q. - Wesley Ivan Hurt, Aug 10 2020

Links

Table of n, a(n) for n=1..86.

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=2..n-1} c(i-1) * c(2*n-i-1) * c(i+1) * c(2*n-i+1), where c is the prime characteristic (A010051).

a(n) = Sum_{i=3..n} c(i) * c(2*n-i) * c(i-2) * c(2*n-i+2), where c is the prime characteristic (A010051). - Wesley Ivan Hurt, Aug 10 2020

Example

a(5) = 1; 2*5 = 10 has two Goldbach partitions, (p,q) = (3,7) and (s,t) = (5,5) where 3,5 are twin primes and 5,7 are twin primes. This makes exactly 1 such pair.

Mathematica

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

Crossrefs

Cf. A002375, A010051.

Sequence in context: A216513 A364419 A291437 * A228101 A078128 A191269

Adjacent sequences: A302228 A302229 A302230 * A302232 A302233 A302234

Keyword

nonn

Author

Wesley Ivan Hurt, Apr 03 2018

Status

approved