A338215 a(n) = A095117(A062298(n)).

1, 1, 1, 3, 3, 5, 5, 6, 8, 9, 9, 11, 11, 12, 13, 14, 14, 16, 16, 17, 19, 20, 20, 21, 22, 24, 25, 27, 27, 28, 28, 29, 30, 32, 33, 34, 34, 35, 36, 37, 37, 39, 39, 40, 42, 43, 43, 44, 45, 46, 47, 49, 49, 50, 51, 52, 54, 55, 55, 57, 57, 58, 59, 60, 62, 63, 63, 64

Offset: 1

Author

Ya-Ping Lu, Oct 17 2020

Keywords

• nonn

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

Crossrefs

Cf. A000720, A062298, A095117, A337978.

Programs

Mathematica

Array[PrimePi[#] + # &[# - PrimePi[#]] &, 68] (* Michael De Vlieger, Nov 04 2020 *)

Python

from sympy import primepi
for n in range(1, 10001):
    b = n - primepi(n)
    a = b + primepi(b)
    print(a)

Formula

a(n) = A095117(A062298(n));

a(n) = n - pi(n) + pi(n - pi(n)), where pi(n) is the prime count of n.