A226237 Sum of the parts in the Goldbach partitions of 2n.

0, 4, 6, 8, 20, 12, 28, 32, 36, 40, 66, 72, 78, 56, 90, 64, 136, 144, 76, 120, 168, 132, 184, 240, 200, 156, 270, 168, 232, 360, 186, 320, 396, 136, 350, 432, 370, 380, 546, 320, 410, 672, 430, 352, 810, 368, 470, 672, 294, 600, 816, 520, 636, 864, 660, 784

Offset: 1

Author

Wesley Ivan Hurt, Aug 25 2013

Keywords

• nonn
• easy

Comments

Goldbach's Conjecture states that every positive even integer > 4 is expressible as the sum of two odd primes in at least one way. This is logically equivalent to the statement that a(n) > 0 for n > 2.

The sum of the parts in the partitions of 2n into exactly two prime parts.

Examples

			a(13) = 78.  Since 2*13 = 26 has exactly 3 Goldbach partitions: (23,3),(19,7), and (13,13).  The sum of the parts gives: 23+19+13+13+7+3 = 78.
		

Links

• 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

Crossrefs

Cf. A045917, A185297, A187129, A187619 (Sum of differences).

Programs

Maple

with(numtheory); A226237:=n->2*n*sum( (pi(i)-pi(i-1)) * (pi(2*n-i)-pi(2*n-i-1)), i=1..n); seq(A226237(n), n=1..100);

Mathematica

Table[ 2 n*Sum[ Floor[2/PrimeOmega[2 n*i - i^2]], {i, 2, n}], {n,
  100}]

Formula

a(n) = 2n * A045917(n). a(n) = A185297(n) + A187129(n), n>1.