A339085 Number of primes p with n - pi(n) < p <= n + pi(n), where pi(n) is the number of primes <= n.

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, 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, 8, 9, 9, 9, 9, 8, 8, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10

Offset

1,2

Comments

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).

Links

Table of n, a(n) for n=1..85.

Ya-Ping Lu, Lower Bounds for the Number of Primes in Some Integer Intervals

Formula

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

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

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

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).

Mathematica

Table[PrimePi[n+PrimePi[n]]-PrimePi[n-PrimePi[n]], {n, 85}] (* Stefano Spezia, Nov 24 2020 *)

Prog

(Python)
from sympy import primepi
for n in range(1, 101):
    m = primepi(n)
    print (primepi(n + m) - primepi(n - m))

Crossrefs

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

Sequence in context: A105528 A076225 A215776 * A140887 A132423 A071995

Adjacent sequences: A339082 A339083 A339084 * A339086 A339087 A339088

Keyword

nonn

Author

Ya-Ping Lu, Nov 23 2020

Status

approved