A334171 Maximum number of rectangles with prime side lengths that can be made with up to 2 perpendicular cuts from all p X q rectangles (p, q prime) such that p + q = 2n.

0, 1, 1, 2, 6, 4, 5, 4, 6, 5, 5, 8, 9, 3, 6, 5, 6, 10, 8, 3, 8, 7, 6, 11, 11, 4, 9, 7, 5, 11, 9, 7, 13, 6, 5, 12, 12, 7, 14, 12, 5, 14, 13, 5, 15, 11, 6, 11, 6, 6, 15, 12, 8, 14, 13, 9, 20, 16, 7, 20, 14, 5, 16, 8, 7, 17, 13, 6, 11, 15, 10, 21, 17, 5, 19, 11, 10

Offset

1,4

Links

Table of n, a(n) for n=1..77.

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=1..n} c(i) * c(2*n-i) * ((1 - c(i-2))*(1 - c(2*n-i-2)) + 2*(1 - [c(i-2) = c(2*n-i-2)]) + 4*c(i-2)*c(2*n-i-2), where [] is the Iverson bracket and c is the prime characteristic (A010051).

Example

a(5) = 6; 2*5 = 10 has 2 Goldbach partitions, (7,3) and (5,5), so we start with two rectangles of size 7 X 3 and 5 X 5. Note that the 7 X 3 rectangle can be cut into two rectangles of size 5 X 3 and 3 X 2 (both prime), but since the cuts must be perpendicular to each other (and not parallel), no smaller rectangles with prime side lengths can be made. Furthermore, the 5 X 5 rectangle can be cut into 4 rectangles with prime side lengths using two perpendicular cuts: 3 X 3, 3 X 2, 2 X 3, and 2 X 2. This makes 6 total rectangles.

Mathematica

Table[Sum[((1 - PrimePi[i - 2] + PrimePi[i - 3]) (1 - PrimePi[2 n - i - 2] + PrimePi[2 n - i - 3]) + 2 (1 - KroneckerDelta[PrimePi[i - 2] - PrimePi[i - 3], PrimePi[2 n - i - 2] - PrimePi[2 n - i - 3]]) + 4 (PrimePi[i - 2] - PrimePi[i - 3]) (PrimePi[2 n - i - 2] - PrimePi[2 n - i - 3])) (PrimePi[i] - PrimePi[i - 1]) (PrimePi[2 n - i] - PrimePi[2 n - i - 1]), {i, n}], {n, 100}]

Crossrefs

Cf. A010051, A045917.

Sequence in context: A195730 A222236 A351001 * A175386 A021382 A356309

Adjacent sequences: A334168 A334169 A334170 * A334172 A334173 A334174

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Apr 17 2020

Status

approved