A343758 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, 4, 6, 17, 22, 27, 32, 62, 72, 82, 92, 151, 168, 185, 202, 219, 236, 253, 270, 408, 436, 464, 492, 689, 730, 771, 812, 853, 894, 935, 976, 1306, 1364, 1422, 1480, 1899, 1976, 2053, 2130, 2207, 2284, 2361, 2438, 3044, 3144, 3244, 3344, 3444, 3544, 3644, 3744, 3844

Offset

1,4

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

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

Example

a(6) = 17; the rectangles are 2 X 4 and 3 X 3. The total area of both rectangles is then 2*4 + 3*3 = 8 + 9 = 17.

Maple

P:= select(isprime, [2, seq(i, i=3..50, 2)]):
f:= proc(n) local p, m, i;
m:= ListTools:-BinaryPlace(P, (n+1)/2);
add(P[i]*(n-P[i]), i=1..m)
end proc:
map(f, [$1..100]); # Robert Israel, Dec 21 2022

Mathematica

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

Crossrefs

Cf. A010051, A343759.

Sequence in context: A226631 A226634 A105271 * A024305 A342231 A320245

Adjacent sequences: A343755 A343756 A343757 * A343759 A343760 A343761

Keyword

nonn

Author

Wesley Ivan Hurt, Apr 27 2021

Status

approved