Question: How to find a sequence odd square free numbers having a property of the Carmichael numbers?

As is well known, a number n is a Charmichael number if and only if for every prime factor p of n, p-1 divides n-1. 

I would like to find a way to identify the following: Odd square free composite numbers n, having at least one prime factor p, with the property that p-1 divides n-1.

obviously the Carmichael numbers are a sub sequence of this. I have already managed to write a code to identify odd square free numbers divisible by the sum of their prime divisors and am interested to see how these data will differ from those of  this new sequence. 

My problem here is to find, given an odd square free number, a way to select and test each prime divisor for the above divisibility requirement. I hope someone can help, thanks in advance

Best regards

David Sycamore.