A337788 The number of primes between n exclusive and n+primepi(n) inclusive.

0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 4, 4, 5, 5, 4, 4, 4, 4, 4, 4, 5, 5, 4, 4, 4, 5, 4, 4, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 4, 5, 5, 6, 6

Offset: 1

Author

Ya-Ping Lu, Oct 27 2020

Keywords

• nonn

Comments

There is at least one prime number in the range of (n, n + primepi(n)], or a(n) >= 1, for n >= 2 (see Corollary 1 in the paper by Ya-Ping Lu attached in the links).

See also the Panaitopol link. - Charles R Greathouse IV, Jul 12 2024

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Ya-Ping Lu, Lower Bounds for the Number of Primes in Some Integer Intervals
• Laurențiu Panaitopol, Intervals containing prime numbers, NNTDM 7 (2001), 4, pp. 111-114.

Crossrefs

Cf. A000720, A095117.

Programs

Mathematica

Table[Count[Range[n+1,n+PrimePi[n]],?PrimeQ],{n,90}] (* _Harvey P. Dale, Aug 28 2024 *)

PARI

a(n) = primepi(n+primepi(n)) - primepi(n); \\ Michel Marcus, Oct 27 2020

Python

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

Formula

a(n) = primepi(n + primepi(n)) - primepi(n)

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

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