cp's OEIS Frontend

A337334 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.

%I A337334 #18 Mar 11 2023 07:55:30
%S A337334 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 A337334 158,181,205,233,266,298,337,378,429,480,539,602,674,751,838,930,1031,
%U A337334 1147,1274,1402,1556,1715,1896,2090,2296,2527,2777,3047,3340,3669,4016
%N A337334 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 A337334 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 A337334 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 A337334 a(n) = pi(b(n)), where b(n) = a(n-1) + b(n-1) with a(0) = b(0) = 1.
%F A337334 a(n) = A000720(A061535(n)), n>=1. - _R. J. Mathar_, Jun 18 2021
%e A337334 a(1) = pi(b(1)) = pi(a(0) + b(0)) = pi(1 + 1) = pi(2) = 1
%e A337334 a(2) = pi(b(2)) = pi(a(1) + b(1)) = pi(1 + 2) = pi(3) = 2
%e A337334 a(3) = pi(b(3)) = pi(a(2) + b(2)) = pi(2 + 3) = pi(5) = 3
%e A337334 a(4) = pi(b(4)) = pi(a(3) + b(3)) = pi(3 + 5) = pi(8) = 4
%e A337334 a(54)= pi(b(54))= pi(a(53)+ b(53))= pi(3669+34327)=pi(37996)=4016
%p A337334 A337334 := proc(n)
%p A337334     option remember;
%p A337334     if n = 0 then
%p A337334         1;
%p A337334     else
%p A337334         numtheory[pi](A061535(n)) ;
%p A337334     end if;
%p A337334 end proc:
%p A337334 seq(A337334(n),n=0..20) ; # _R. J. Mathar_, Jun 18 2021
%o A337334 (Python)
%o A337334 from sympy import primepi
%o A337334 a_last = 1
%o A337334 b_last = 1
%o A337334 for n in range(1, 1001):
%o A337334     b = a_last + b_last
%o A337334     a = primepi(b)
%o A337334     print(a)
%o A337334     a_last = a
%o A337334     b_last = b
%Y A337334 Cf. A000720 (pi), A014688 (prime(n)+n), A332086.
%K A337334 nonn
%O A337334 0,3
%A A337334 _Ya-Ping Lu_, Aug 23 2020
%E A337334 a(0) inserted by _R. J. Mathar_, Jun 18 2021