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

1,2

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.

Links

Table of n, a(n) for n=1..56.

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

Example

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.

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}]

Crossrefs

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

Sequence in context: A346868 A279896 A247280 * A114315 A058238 A191697

Adjacent sequences: A226234 A226235 A226236 * A226238 A226239 A226240

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Aug 25 2013

Status

approved