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

Author

Wesley Ivan Hurt, Oct 28 2017

Keywords

• nonn
• easy

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).

Examples

			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.
		

Links

• 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. A010051, A166081, A243485.

Programs

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 *)

PARI

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

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