A339085 - OEIS

%I #8 Nov 26 2020 23:06:40

%S 0,2,3,2,3,2,3,3,3,3,3,3,4,4,4,4,5,4,4,4,4,4,5,5,5,4,4,4,4,4,5,6,6,5,

%T 5,6,6,6,6,6,7,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,7,8,8,8,8,8,8,8,8,

%U 8,9,9,9,9,8,8,9,9,9,10,10,10,10,10,10,10

%N Number of primes p with n - pi(n) < p <= n + pi(n), where pi(n) is the number of primes <= n.

%C a(n) >= 2 if n >= 2 and a(n) >= 3 if n is a prime >= 3 (see the paper by Ya-Ping Lu attached in the links).

%H Ya-Ping Lu, <a href="/A337788/a337788.pdf">Lower Bounds for the Number of Primes in Some Integer Intervals</a>

%F a(n) = pi(n + pi(n)) - pi(n - pi(n)).

%F a(n) = A000720(n + A000720(n)) - A000720(n - A000720(n)).

%F a(n) = A000720(A095117(n)) - A000720(A062298(n)).

%F a(n) = A337788(n) + A338521(n) + isprime(n), where isprime(n) = 1 (if n is a prime) or 0 (if n is not a prime).

%t Table[PrimePi[n+PrimePi[n]]-PrimePi[n-PrimePi[n]],{n,85}] (* _Stefano Spezia_, Nov 24 2020 *)

%o (Python)

%o from sympy import primepi

%o for n in range(1, 101):

%o     m = primepi(n)

%o     print (primepi(n + m) - primepi(n - m))

%Y Cf. A000720, A095117, A062298, A337788, A338521.

%K nonn

%O 1,2

%A _Ya-Ping Lu_, Nov 23 2020