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Total area of all p X r rectangles where n = p + r, p < r, p is prime and r is a positive integer.
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%I #4 Apr 27 2021 19:07:52

%S 0,0,0,0,6,8,22,27,32,37,72,82,92,102,168,185,202,219,236,253,270,287,

%T 436,464,492,520,730,771,812,853,894,935,976,1017,1364,1422,1480,1538,

%U 1976,2053,2130,2207,2284,2361,2438,2515,3144,3244,3344,3444,3544,3644,3744,3844,3944

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

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

%e 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.

%e 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.

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

%Y Cf. A010051, A343758 (non-distinct).

%K nonn

%O 1,5

%A _Wesley Ivan Hurt_, Apr 27 2021