A352749 a(n) = pi(n) * (pi(2n-1) - pi(n-1)).

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, 54, 63, 63, 70, 70, 88, 77, 77, 88, 88, 99, 120, 108, 108, 120, 130, 130, 140, 126, 140, 140, 150, 135, 150, 150, 165, 180, 192, 192, 208, 208, 224, 224, 238, 221, 234, 216, 216, 234

Offset: 1

Author

Wesley Ivan Hurt, Apr 01 2022

Keywords

• nonn
• easy

Comments

Number of ordered pairs of prime numbers, (p,q), such that p <= n <= q < 2n.

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.

Examples

			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).
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.
		

Links

• Eric Weisstein's World of Mathematics, Prime Counting Function.
• Wikipedia, Prime-counting function.

Crossrefs

Cf. A000720 (pi), A035250, A352753, A352754.

Programs

Mathematica

Table[PrimePi[n] (PrimePi[2 n - 1] - PrimePi[n - 1]), {n, 100}]

PARI

a(n) = primepi(n)*(primepi(2*n-1) - primepi(n-1)); \\ Michel Marcus, Apr 01 2022

Formula

a(n) = Sum_{p <= n <= q < 2n, p,q prime} 1.

a(n) = A000720(n) * A035250(n). - Bernard Schott, Apr 02 2022