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A226237 Sum of the parts in the Goldbach partitions of 2n. 4
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 (list; graph; refs; listen; history; text; internal format)
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: A210459 A279896 A247280 * A114315 A058238 A309283

Adjacent sequences:  A226234 A226235 A226236 * A226238 A226239 A226240

KEYWORD

nonn,easy

AUTHOR

Wesley Ivan Hurt, Aug 25 2013

STATUS

approved

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Last modified February 20 04:12 EST 2020. Contains 332063 sequences. (Running on oeis4.)