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%I #27 Sep 11 2020 11:24:40
%S 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,
%T 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,
%U 10,11,11,11,11,11,11,12,12,12,12,13,13,13,13
%N a(n) = pi(prime(n) + n) - n, where pi is the prime counting function.
%C 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.
%H Robert Israel, <a href="/A332086/b332086.txt">Table of n, a(n) for n = 1..10000</a>
%H Ya-Ping Lu and Shu-Fang Deng, <a href="https://arxiv.org/abs/2007.15282">An upper bound for the prime gap</a>, arXiv:2007.15282 [math.GM], 2020.
%F a(n) = pi(prime(n) + n) - n.
%F a(n) = A000720(A014688(n)) - n. - _Michel Marcus_, Aug 23 2020
%e a(1) = pi(p_1 + 1) - 1 = pi(2 + 1) - 1 = 2 - 1 = 1;
%e a(2) = pi(p_2 + 2) - 2 = pi(3 + 2) - 2 = 3 - 2 = 1;
%e a(6) = pi(p_6 + 6) - 6 = pi(13 + 6) - 6 = 8 - 6 = 2;
%e a(80) = pi(p_80 + 80) - 80 = pi(409 + 80) - 80 = 93 - 80 = 13.
%p f:= n -> numtheory:-pi(ithprime(n)+n)-n:
%p map(f, [$1..100]); # _Robert Israel_, Sep 08 2020
%t a[n_] := PrimePi[Prime[n] + n] - n; Array[a, 100] (* _Amiram Eldar_, Aug 23 2020 *)
%o (Python)
%o from sympy import prime, primepi
%o for n in range(1, 1001):
%o a = primepi(prime(n) + n) - n
%o print(a)
%o (PARI) a(n) = primepi(prime(n) + n) - n; \\ _Michel Marcus_, Aug 23 2020
%Y Cf. A000720 (pi), A014688 (prime(n)+n).
%K nonn
%O 1,6
%A _Ya-Ping Lu_, Aug 22 2020
%E Name edited by _Michel Marcus_, Sep 02 2020
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