OFFSET
1,4
COMMENTS
It can be shown that there is at least one prime number between n-pi(n) and n for n >= 3, or pi(n-1)-pi(n-pi(n)) >= 1. Since a(n)=n-pi(n)+pi(n-pi(n)) <= n-pi(n-1)+pi(n-pi(n)) <= n-1, we have a(n) < n for n > 1.
a(n)-a(n-1) = 1 - (pi(n)-pi(n-1)) + pi(n-pi(n)) - pi(n-(1+pi(n-1))), where pi(n)-pi(n-1) <= 1 and 1+pi(n-1) >= pi(n) or pi(n-(1+pi(n-1))) <= pi(n-pi(n)). Thus, a(n) - a(n-1) >= 0, meaning that this is a nondecreasing sequence.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10000
FORMULA
MATHEMATICA
Array[PrimePi[#] + # &[# - PrimePi[#]] &, 68] (* Michael De Vlieger, Nov 04 2020 *)
PROG
(Python)
from sympy import primepi
for n in range(1, 10001):
b = n - primepi(n)
a = b + primepi(b)
print(a)
CROSSREFS
KEYWORD
nonn
AUTHOR
Ya-Ping Lu, Oct 17 2020
STATUS
approved