

A243485


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


3



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
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OFFSET

1,4


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


FORMULA

a(n) = Sum_{i=2..n/2} i*(ni) * A064911(i*(ni)).
a(n) = Sum_{i=1..floor(n/2)} i * (ni) * A010051(i) * A010051(ni).  Wesley Ivan Hurt, Oct 29 2017


MAPLE

with(numtheory): A243485:=n>add(i*(ni)*(pi(i)pi(i1))*(pi(ni)pi(ni1)), 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}]


CROSSREFS

Cf. A002372, A002375, A045917, A064911, A117929.
Sequence in context: A189553 A189482 A099303 * A310665 A005659 A010462
Adjacent sequences: A243482 A243483 A243484 * A243486 A243487 A243488


KEYWORD

nonn,easy


AUTHOR

Wesley Ivan Hurt, Jun 05 2014


STATUS

approved



