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%I #14 Apr 02 2022 18:35:25
%S 0,2,4,4,6,6,12,8,12,16,20,20,24,18,24,30,35,28,40,32,40,48,54,54,54,
%T 54,63,63,70,70,88,77,77,88,88,99,120,108,108,120,130,130,140,126,140,
%U 140,150,135,150,150,165,180,192,192,208,208,224,224,238,221,234,216,216,234
%N a(n) = pi(n) * (pi(2n-1) - pi(n-1)).
%C Number of ordered pairs of prime numbers, (p,q), such that p <= n <= q < 2n.
%C Also the number of ordered pairs of prime numbers, (p,q) that can be made with p <= q, where p and q appear as the smaller and larger parts (respectively) of the partitions of 2n into 2 parts that contain at least 1 prime.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeCountingFunction.html">Prime Counting Function</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Prime-counting_function">Prime-counting function</a>.
%F a(n) = Sum_{p <= n <= q < 2n, p,q prime} 1.
%F a(n) = A000720(n) * A035250(n). - _Bernard Schott_, Apr 02 2022
%e a(5) = 6; 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).
%e Another interpretation for a(5): the 3 partitions of 2*5 = 10 into 2 parts containing at least one prime are 2+8 = 3+7 = 5+5. There are 6 ordered pairs of primes (p,q) that can be made with p <= q, which are the same ordered pairs in the previous example.
%t Table[PrimePi[n] (PrimePi[2 n - 1] - PrimePi[n - 1]), {n, 100}]
%o (PARI) a(n) = primepi(n)*(primepi(2*n-1) - primepi(n-1)); \\ _Michel Marcus_, Apr 01 2022
%Y Cf. A000720 (pi), A035250, A352753, A352754.
%K nonn,easy
%O 1,2
%A _Wesley Ivan Hurt_, Apr 01 2022
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