A294345 Sum of the products of the smaller and larger parts of the Goldbach partitions of n into two distinct parts.

0, 0, 0, 0, 6, 0, 10, 15, 14, 21, 0, 35, 22, 33, 26, 94, 0, 142, 34, 142, 38, 142, 0, 357, 46, 202, 0, 302, 0, 591, 58, 334, 62, 491, 0, 980, 0, 217, 74, 821, 0, 1340, 82, 785, 86, 827, 0, 1987, 94, 1512, 0, 1353, 0, 2677, 106, 1421, 0, 1479, 0, 4242, 118

Offset

1,5

Comments

Sum of the areas of the distinct rectangles with prime length and width such that L + W = n, W < L. For example, a(14) = 33; the only rectangle is 3 X 11 and 3*11 = 33 (the 7 X 7 rectangle is not considered since we have W < L).

Links

Table of n, a(n) for n=1..61.

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..floor((n-1)/2)} i * (n-i) * c(i) * c(n-i), where c is the prime characteristic (A010051).

a(k) = 0, for k in A166081. - Michel Marcus, Oct 30 2017

Example

a(22) = 142; the Goldbach partitions of 22 are (19,3), (17,5) and (11,11) (we do not consider (11,11) since we only count prime parts which are distinct). Then the sum of the products of the smaller and larger parts from each partition is 19*3 + 17*5 = 142.

Maple

with(numtheory): A294345:=n->add(i*(n-i)*(pi(i)-pi(i-1))*(pi(n-i)-pi(n-i-1)), i=2..floor((n-1)/2)): seq(A294345(n), n=1..100);

Mathematica

Table[Sum[i (n - i) (PrimePi[i] - PrimePi[i - 1]) (PrimePi[n - i] - PrimePi[n - i - 1]), {i, 2, Floor[(n-1)/2]}], {n, 60}]
Table[Total[Times@@@Select[IntegerPartitions[n, {2}], AllTrue[#, PrimeQ] && #[[1]]!=#[[2]]&]], {n, 70}] (* Harvey P. Dale, Jul 29 2021 *)

Prog

(PARI) a(n) = sum(i=1, (n-1)\2, i*isprime(i)*(n-i)*isprime(n-i)); \\ Michel Marcus, Nov 08 2017

Crossrefs

Cf. A010051, A166081, A243485.

Sequence in context: A187429 A083573 A117006 * A073764 A158897 A033458

Adjacent sequences: A294342 A294343 A294344 * A294346 A294347 A294348

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Oct 28 2017

Status

approved