A126706 Positive integers which are neither squarefree integers nor prime powers.

12, 18, 20, 24, 28, 36, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 72, 75, 76, 80, 84, 88, 90, 92, 96, 98, 99, 100, 104, 108, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 180, 184, 188

Offset: 1

Author

Leroy Quet, Feb 11 2007

Keywords

• nonn

Examples

			45 is in the sequence because 45=3^2*5, i.e., neither squarefree nor a prime power.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A005117, A000961, A059404.

Programs

Maple

with(numtheory): a:=proc(n) if mobius(n)=0 and nops(factorset(n))>1 then n else fi end: seq(a(n), n=1..230); # Emeric Deutsch, Feb 17 2007

Mathematica

Select[Range[200], Max @@ Last /@ FactorInteger[ # ] >1 && Length[FactorInteger[ # ]] > 1 &] (* Ray Chandler, Feb 17 2007 *)
Select[Range[200],!SquareFreeQ[#]&&!PrimePowerQ[#]&] (* Harvey P. Dale, Aug 05 2023 *)

PARI

isok(k) = !issquarefree(k) && !isprimepower(k); \\ Michel Marcus, Nov 02 2022

Python

from math import isqrt
from sympy import primepi, integer_nthroot, mobius
def A126706(n):
    def f(x): return int(n+sum(primepi(integer_nthroot(x,k)[0]) for k in range(2,x.bit_length()))+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 m # Chai Wah Wu, Aug 15 2024

Extensions

Extended by Emeric Deutsch and Ray Chandler, Feb 17 2007