What Is a Composite Number With No Co-Prime Factors?

[Ventrella]

I have come across a question in my explorations of recursive tilings using Gaussian and Eisenstein Integers. It seems there might be a class of composite numbers which have no factors (divisors) that are co-prime. So, for instance: 10 is composite, but its factors (2 and 5) are co-prime. The same is true for 6 and 12. However, 4, 8, 9, 16, 25 do not have factors which are co-prime. (I am including composite numbers with only one factor in this list). These are then examples of the class of numbers I am looking to identify.

Without going into the details and reasoning for asking this question (unless it helps in finding the answer), I am initially curious if such a class of composites have been identified, and if so, perhaps there may be some significance that can shed some light on my problem.

Any insights, thoughts?

Thanks!
-Jeffrey

 

[Stephen Tashi]

Perhaps you want "prime power" numbers: http://en.wikipedia.org/wiki/Prime_power

 

[Ventrella]

Thanks Stephen,

I think your suggestion is helping me to converge on the answer, although I clearly said that I am looking at composite numbers only. However I think that if I take the series pointed out (sequence A246655 in OEIS), and strip out all the primes, I will have the set I am looking for :)

My intuition tells me that this is right because fractals (as I am exploring them with tilings in lattices) obey power laws.

Thanks!
-j