Composite numbers (p*q*r*…)^k where p, q, r,… are distinct primes and k is any positive integer.

1, 4, 6, 8, 9, 10, 14, 15, 16, 21, 22, 25, 26, 27, 30, 32, 33, 34, 35, 36, 38, 39, 42, 46, 49, 51, 55, 57, 58, 62, 64, 65, 66, 69, 70, 74, 77, 78, 81, 82, 85, 86, 87, 91, 93, 94, 95, 100, 102, 105, 106, 110, 111, 114, 115, 118, 119, 121, 122, 123, 125, 128, 129

1

Because these are composite, we must have k>1 or at least two different primes in the product p*q*r*…. The number 1 is included. These are the numbers counted in S000570. These are similar to perfect powers, but also include numbers to the first power. What is the density of these numbers?

T. D. Noe, Plot of 1000 terms

T. D. Noe, Table of 1000 terms

Eric W. Weisstein, MathWorld: Perfect Power

(Mma) Select[Range[200], ! PrimeQ[#] && Length[Union[Transpose[FactorInteger[#]][[2]]]] == 1 &]

nonn

T. D. Noe, Apr 10 2015