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%I #5 Apr 02 2022 18:36:09
%S 0,9,26,34,56,74,175,130,215,308,412,472,596,477,692,919,1123,946,
%T 1497,1268,1673,2094,2436,2652,2652,2652,3229,3229,3713,4013,5372,
%U 4871,4871,5768,5768,6709,8594,7953,7953,9098,10102,10648,11714,10831,12358,12358,13510
%N a(n) = pi(n) * (Sum_{n <= q < 2n, q prime} q) + (pi(2n-1) - pi(n-1)) * (Sum_{p <= n, p prime} p).
%C Sum of all the parts from all ordered pairs of prime numbers, (p,q), such that p <= n <= q < 2n.
%F a(n) = A352753(n) + A352754(n).
%e a(5) = 56; there are 6 ordered pairs of prime numbers, (p,q), such that p <= 5 <= q < 10: (2,5), (2,7), (3,5), (3,7), (5,5), and (5,7). The sum of all the parts gives 2+5+2+7+3+5+3+7+5+5+5+7 = 56.
%t Table[Sum[Sum[k (PrimePi[k] - PrimePi[k - 1]) (PrimePi[2 n - i] - PrimePi[2 n - i - 1]), {k, n}], {i, n}] + PrimePi[n] Sum[(2 n - k) (PrimePi[2 n - k] - PrimePi[2 n - k - 1]), {k, n}], {n, 100}]
%Y Cf. A000720 (pi), A073837, A352749, A352753, A352754, A352777.
%K nonn
%O 1,2
%A _Wesley Ivan Hurt_, Apr 02 2022
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