A303946 Numbers that are neither squarefree nor perfect powers.

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 72, 75, 76, 80, 84, 88, 90, 92, 96, 98, 99, 104, 108, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189

Offset: 1

Author

Gus Wiseman, May 03 2018

Keywords

• nonn

Comments

First differs from A059404 at a(40) = 147, A059404(40) = 144.

First differs from A126706 at a(6) = 40, A126706(6) = 36.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000009, A000837, A001597, A005117, A007916, A013929, A047966, A052409, A052410, A072774, A073576, A126706, A132350.

Programs

Maple

filter:= proc(n) local F;
F:= map(t->t[2],ifactors(n)[2]);
max(F)>1 and igcd(op(F))=1
end proc:
select(filter, [$1..1000]); # Robert Israel, May 06 2018

Mathematica

Select[Range[200], !SquareFreeQ[#] && GCD@@FactorInteger[#][[All, 2]] == 1 &]

PARI

isok(n) = !issquarefree(n) && !ispower(n); \\ Michel Marcus, May 05 2018

Python

from math import isqrt
from sympy import mobius, integer_nthroot
def A303946(n):
    def f(x): return int(n+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

a(n) ~ n/k, where k = 1 - 1/zeta(2) = 1 - 6/Pi^2 = A229099. - Charles R Greathouse IV, Jun 01 2018