OFFSET
0,3
COMMENTS
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)).
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(b(n)), where b(n) = a(n-1) + b(n-1) with a(0) = b(0) = 1.
EXAMPLE
a(1) = pi(b(1)) = pi(a(0) + b(0)) = pi(1 + 1) = pi(2) = 1
a(2) = pi(b(2)) = pi(a(1) + b(1)) = pi(1 + 2) = pi(3) = 2
a(3) = pi(b(3)) = pi(a(2) + b(2)) = pi(2 + 3) = pi(5) = 3
a(4) = pi(b(4)) = pi(a(3) + b(3)) = pi(3 + 5) = pi(8) = 4
a(54)= pi(b(54))= pi(a(53)+ b(53))= pi(3669+34327)=pi(37996)=4016
MAPLE
A337334 := proc(n)
option remember;
if n = 0 then
1;
else
numtheory[pi](A061535(n)) ;
end if;
end proc:
seq(A337334(n), n=0..20) ; # R. J. Mathar, Jun 18 2021
PROG
(Python)
from sympy import primepi
a_last = 1
b_last = 1
for n in range(1, 1001):
b = a_last + b_last
a = primepi(b)
print(a)
a_last = a
b_last = b
CROSSREFS
KEYWORD
nonn
AUTHOR
Ya-Ping Lu, Aug 23 2020
EXTENSIONS
a(0) inserted by R. J. Mathar, Jun 18 2021
STATUS
approved