%I #27 Dec 14 2025 10:17:38

%S 1,1,2,3,4,5,7,9,11,14,16,21,24,30,35,42,48,58,67,78,91,103,121,138,

%T 158,181,205,233,266,298,337,378,429,480,539,602,674,751,838,930,1031,

%U 1147,1274,1402,1556,1715,1896,2090,2296,2527,2777,3047,3340,3669,4016

%N a(n) = pi(b(n)), where pi is the prime counting function (A000720) and b(n) = a(n-1) + b(n-1) with a(0) = b(0) = 1.

%C It can be proved that this is an increasing sequence from the theorem of Lu and Deng (see LINKS), which states "the prime gap of a prime number is less than or equal to the prime count of the prime number", or prime(n+1) - prime(n) <= pi(prime(n)).

%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(b(n)), where b(n) = a(n-1) + b(n-1) with a(0) = b(0) = 1.

%F a(n) = A000720(A061535(n)), n>=1. - _R. J. Mathar_, Jun 18 2021

%e a(1) = pi(b(1)) = pi(a(0) + b(0)) = pi(1 + 1) = pi(2) = 1.

%e a(2) = pi(b(2)) = pi(a(1) + b(1)) = pi(1 + 2) = pi(3) = 2.

%e a(3) = pi(b(3)) = pi(a(2) + b(2)) = pi(2 + 3) = pi(5) = 3.

%e a(4) = pi(b(4)) = pi(a(3) + b(3)) = pi(3 + 5) = pi(8) = 4.

%e a(54) = pi(b(54)) = pi(a(53) + b(53)) = pi(3669 + 34327) = pi(37996) = 4016.

%p A337334 := proc(n)

%p option remember;

%p if n = 0 then

%p 1;

%p else

%p numtheory[pi](A061535(n)) ;

%p end if;

%p end proc:

%p seq(A337334(n),n=0..20) ; # _R. J. Mathar_, Jun 18 2021

%t Join[{1}, Differences[NestList[# + PrimePi[#] &, 2, 50]]] (* _Paolo Xausa_, Dec 13 2025 *)

%o (Python)

%o from sympy import primepi

%o a_last = 1

%o b_last = 1

%o for n in range(1, 1001):

%o b = a_last + b_last

%o a = primepi(b)

%o print(a)

%o a_last = a

%o b_last = b

%Y Cf. A000720 (pi), A014688 (prime(n)+n), A332086.

%Y From n >= 1, first differences of A061535.

%K nonn

%O 0,3

%A _Ya-Ping Lu_, Aug 23 2020

%E a(0) inserted by _R. J. Mathar_, Jun 18 2021