A243485 Sum of all the products formed by multiplying the corresponding smaller and larger parts of the Goldbach partitions of n.

0, 0, 0, 4, 6, 9, 10, 15, 14, 46, 0, 35, 22, 82, 26, 94, 0, 142, 34, 142, 38, 263, 0, 357, 46, 371, 0, 302, 0, 591, 58, 334, 62, 780, 0, 980, 0, 578, 74, 821, 0, 1340, 82, 785, 86, 1356, 0, 1987, 94, 1512, 0, 1353, 0, 2677, 106, 1421, 0, 2320, 0, 4242, 118

Offset: 1

Author

Wesley Ivan Hurt, Jun 05 2014

Keywords

• nonn
• easy

Comments

a(n) is even for odd n.

If Goldbach's conjecture is true, a(n) > 0 for all even n > 2.

Sum of the areas of the distinct rectangles with prime length and width such that L + W = n, W <= L. For example, a(16) = 94; the two rectangles are 3 X 13 and 5 X 11, and the sum of their areas is 3*13 + 5*11 = 94. - Wesley Ivan Hurt, Oct 28 2017

Links

• 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

Crossrefs

Cf. A002372, A002375, A045917, A064911, A117929.

Programs

Maple

with(numtheory): A243485:=n->add(i*(n-i)*(pi(i)-pi(i-1))*(pi(n-i)-pi(n-i-1)), i=1..floor(n/2)): seq(A243485(n), n=1..100); # Wesley Ivan Hurt, Oct 29 2017

Mathematica

Table[Sum[i*(n - i)*Floor[2/PrimeOmega[i (n - i)]], {i, 2, n/2}], {n,
  50}]

Formula

a(n) = Sum_{i=2..n/2} i*(n-i) * A064911(i*(n-i)).

a(n) = Sum_{i=1..floor(n/2)} i * (n-i) * A010051(i) * A010051(n-i). - Wesley Ivan Hurt, Oct 29 2017