There's no easy way to explain this, but please bear with me. I'll try to keep it slow and simple.
I'm looking for a property that is related to the generalised pentagonal numbers (A001318 in the OEIS database).
I'm sure you are aware that if you take any number of the form 6*n +/- 1, square it, then minus one, then it is divisible by 24. This set of numbers (6 *n +/- 1) obviously includes all the prime numbers, which is what I'm interested in here.
Before I get to the second bit I am interested in, I need to explain about the properties of the set of numbers generated from 6*n +/- 1, when they are squared, one is taken away and then divided by 24, the generalised pentagonal numbers.
Now pick a prime divisor d, e.g. 7, and work out how many times the generalised pentagonal numbers are divisible by it exactly with no remainder. This works out, in general, as 4 times in a sequence of 2*d numbers on average. Now, this can be proven to occur regularly at evenly spaced intervals, for any given prime divisor, as the gpn numbers increase.
Now onto my question at last. Suppose you have an unknown prime number that is one factor of a semi-prime composite. Is there a sequence of numbers such that when they multiply any given prime number such as this, and the result is squared then one taken away, and it's then divided by 24, that it maintains the same sequence and regularity of intervals at which the generalised pentagonal numbers are divisible by the prime divisor d. In other words is there a sequence of numbers that can multiply a prime successively, and then when the result is squared, one taken away, and it divided by 24, that the properties of the generalised pentagonal numbers are preserved in the sequence that then results, regarding divisors.
P.S. I have taken as a first attempt the sequence 0, 1, 5, 7, 11, 13, and while this maintains the regularity of the resultant numbers being divisible by d, the intervals are different.
I hope I have explained this so it can be understood, but I can provide examples to make it easier to understand if need be.