A243351 Difference between 2n and the n-th squarefree number: a(n) = 2n - A005117(n).

1, 2, 3, 3, 4, 5, 4, 5, 5, 6, 7, 7, 7, 7, 8, 9, 8, 7, 8, 9, 9, 10, 11, 11, 12, 13, 13, 14, 15, 14, 15, 13, 13, 13, 13, 14, 15, 15, 16, 15, 16, 17, 17, 18, 19, 19, 20, 19, 20, 21, 20, 21, 21, 22, 23, 23, 23, 23, 24, 25, 25, 23, 24, 25, 25, 26, 27, 27, 28, 29, 29, 30

Offset: 1

Author

Antti Karttunen, Jun 04 2014

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10001

Crossrefs

Cf. A005117, A152416, A243348.

Programs

Mathematica

Module[{sf=Select[Range[150],SquareFreeQ]},Table[2n-sf[[n]],{n,Length[ sf]}]] (* Harvey P. Dale, Jun 26 2021 *)

Python

from math import isqrt
from sympy import mobius
def A243351(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return (n<<1)-m # Chai Wah Wu, Aug 12 2024

Formula

a(n) = 2n - A005117(n).

a(n) = n - A243348(n).

a(n) ~ c * n, where c = 2 - Pi^2/6 (A152416). - Amiram Eldar, Mar 04 2024