A304449 Numbers that are either squarefree or a perfect power.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89

Offset: 1

Author

Gus Wiseman, May 12 2018

Keywords

• nonn

Comments

First differs from A072774 at a(105) = 144, A072774(105) = 145.

Apparently the 1 and the members of A062770. - R. J. Mathar, May 22 2018

Crossrefs

Cf. A000009, A000720, A000837, A000837, A001222, A001597, A005117, A007916, A013929, A047966, A072774, A073576, A126706, A303554, A303946.

Programs

Mathematica

Select[Range[150],SquareFreeQ[#]||GCD@@FactorInteger[#][[All,2]]>1&]

PARI

isok(n) = issquarefree(n) || ispower(n); \\ Michel Marcus, May 13 2018

Python

from math import isqrt
from sympy import mobius, integer_nthroot
def A304449(n):
    def f(x): return int(n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 19 2024

Formula

Union of A005117 and A001597. Complement of A303946.