A332086 a(n) = pi(prime(n) + n) - n, where pi is the prime counting function.

1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 3, 3, 2, 3, 3, 4, 4, 4, 4, 3, 4, 4, 6, 5, 5, 4, 4, 4, 4, 6, 6, 6, 6, 7, 6, 7, 8, 7, 7, 6, 6, 8, 7, 8, 7, 8, 10, 9, 9, 10, 9, 9, 8, 9, 10, 9, 8, 8, 8, 7, 9, 10, 10, 9, 10, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13

Offset: 1

Author

Ya-Ping Lu, Aug 22 2020

Keywords

• nonn

Comments

This sequence is related to a theorem of Lu and Deng (see LINKS): “The prime gap of a prime number is less than or equal to the prime count of the prime number”, which is equivalent to “There exists at least one prime number between p and p+pi(p)+1”, or pi(p+pi(p)) - pi(p) > 1, where pi is prime counting function. The n-th term of the sequence, a(n), is the number of prime number between the n-th prime number p_n and p_n + pi(p_n) + 1. According to the theorem, a(n) >= 1.

Examples

			a(1) = pi(p_1 + 1) - 1 = pi(2 + 1) - 1 = 2 - 1 = 1;
a(2) = pi(p_2 + 2) - 2 = pi(3 + 2) - 2 = 3 - 2 = 1;
a(6) = pi(p_6 + 6) - 6 = pi(13 + 6) - 6 = 8 - 6 = 2;
a(80) = pi(p_80 + 80) - 80 = pi(409 + 80) - 80 = 93 - 80 = 13.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Ya-Ping Lu and Shu-Fang Deng, An upper bound for the prime gap, arXiv:2007.15282 [math.GM], 2020.

Crossrefs

Cf. A000720 (pi), A014688 (prime(n)+n).

Programs

Maple

f:= n -> numtheory:-pi(ithprime(n)+n)-n:
map(f, [$1..100]); # Robert Israel, Sep 08 2020

Mathematica

a[n_] := PrimePi[Prime[n] + n] - n; Array[a, 100] (* Amiram Eldar, Aug 23 2020 *)

PARI

a(n) = primepi(prime(n) + n) - n; \\ Michel Marcus, Aug 23 2020

Python

from sympy import prime, primepi
for n in range(1, 1001):
    a = primepi(prime(n) + n) - n
    print(a)

Formula

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

a(n) = A000720(A014688(n)) - n. - Michel Marcus, Aug 23 2020

Extensions

Name edited by Michel Marcus, Sep 02 2020