A106543 Composite numbers that are not perfect powers.

6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 102, 104, 105, 106

Offset: 1

Author

Alexandre Wajnberg, May 08 2005

Keywords

• easy
• nonn

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Intersection of A002808 and A007916.

Cf. A105642, A089229.

Programs

Mathematica

perfPQ[n_]:=GCD@@FactorInteger[n][[All,2]]>1; Select[Range[110], CompositeQ[ #] && !perfPQ[#]&] (* Harvey P. Dale, Oct 10 2017 *)

PARI

lista(nn)=forcomposite(i=1, nn, if (! ispower(i), print1(i, ", "));); \\ Michel Marcus, Jun 27 2013

PARI

is(n)=!isprime(n) && !ispower(n) && n>1 \\ Charles R Greathouse IV, Oct 19 2015

Python

from sympy import primepi, mobius, integer_nthroot
def A106543(n):
    def f(x): return int(n+1+primepi(x)-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, Oct 12 2024

SageMath

def A106543_list(n) : return [k for k in (2..n) if not k.is_prime() and not k.is_perfect_power()]
A106543_list(106) # Terry D. Grant, Jul 17 2016

Formula

a(n) = n + O(n/log n). - Charles R Greathouse IV, Oct 03 2011