A343759 Total area of all p X r rectangles where n = p + r, p < r, p is prime and r is a positive integer.

0, 0, 0, 0, 6, 8, 22, 27, 32, 37, 72, 82, 92, 102, 168, 185, 202, 219, 236, 253, 270, 287, 436, 464, 492, 520, 730, 771, 812, 853, 894, 935, 976, 1017, 1364, 1422, 1480, 1538, 1976, 2053, 2130, 2207, 2284, 2361, 2438, 2515, 3144, 3244, 3344, 3444, 3544, 3644, 3744, 3844, 3944

Offset

1,5

Links

Table of n, a(n) for n=1..55.

Formula

a(n) = Sum_{k=1..floor((n-1)/2)} k*(n-k)*c(k), where c is the prime characteristic (A010051).

Example

a(7) = 22; The two rectangles are 2 X 5 and 3 X 4. The total area is then 2*5 + 3*4 = 10 + 12 = 22.
a(10) = 37; The two rectangles are 2 X 8 and 3 X 7 (we don't count 5 X 5 since p < r). The total area is then 2*8 + 3*7 = 16 + 21 = 37.

Mathematica

Table[Sum[i*(n - i) (PrimePi[i] - PrimePi[i - 1]), {i, Floor[(n - 1)/2]}], {n, 60}]

Crossrefs

Cf. A010051, A343758 (non-distinct).

Sequence in context: A084962 A365649 A296354 * A024306 A024868 A262199

Adjacent sequences: A343756 A343757 A343758 * A343760 A343761 A343762

Keyword

nonn

Author

Wesley Ivan Hurt, Apr 27 2021

Status

approved