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A332086
a(n) = pi(prime(n) + n) - n, where pi is the prime counting function.
6
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
1,6
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.
LINKS
Ya-Ping Lu and Shu-Fang Deng, An upper bound for the prime gap, arXiv:2007.15282 [math.GM], 2020.
FORMULA
a(n) = pi(prime(n) + n) - n.
a(n) = A000720(A014688(n)) - n. - Michel Marcus, Aug 23 2020
EXAMPLE
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.
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 *)
PROG
(Python)
from sympy import prime, primepi
for n in range(1, 1001):
a = primepi(prime(n) + n) - n
print(a)
(PARI) a(n) = primepi(prime(n) + n) - n; \\ Michel Marcus, Aug 23 2020
CROSSREFS
Cf. A000720 (pi), A014688 (prime(n)+n).
Sequence in context: A333003 A352072 A365067 * A197081 A029395 A029282
KEYWORD
nonn
AUTHOR
Ya-Ping Lu, Aug 22 2020
EXTENSIONS
Name edited by Michel Marcus, Sep 02 2020
STATUS
approved