A187619 -id:A187619 - cp's OEIS frontend

A279726 Partial sums of A187619.

0, 0, 0, 2, 6, 8, 16, 32, 44, 64, 92, 118, 150, 174, 202, 234, 298, 358, 382, 440, 512, 598, 686, 808, 924, 1002, 1130, 1228, 1336, 1480, 1560, 1762, 1966, 2026, 2210, 2426, 2614, 2840, 3132, 3300, 3496, 3812, 4072, 4240, 4616, 4852, 5068, 5402, 5522, 5826, 6234, 6512

Offset: 1

Wesley Ivan Hurt, Dec 17 2016

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
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

Cf. A010051, A045917.

with(numtheory): a:=n->2*add(add( (pi(i)-pi(i-1)) * (pi(2*k-i)-pi(2*k-i-1)) * (k-i), i=3..k), k=1..n): seq(a(n), n=1..100);

Accumulate@ Table[2 Sum[Boole[PrimeQ@ i] Boole[PrimeQ[2 n - i]] (n - i), {i, 3, n}], {n, 56}] (* Michael De Vlieger, Dec 17 2016 *)

a(n) = 2 * Sum_{k=1..n} (Sum_{i=3..k} A010051(i) * A010051(2k-i) * (k-i)).

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

Original entry on oeis.org

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

Wesley Ivan Hurt, Aug 25 2013

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.

			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.

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

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

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);

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

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

cp's OEIS Frontend

A279726 Partial sums of A187619.

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