A wheel factorization is when you remove all the multiples of primes (up to a prime number P) from the product of all primes up to and including P. Examples:
For P=5, you remove all the multiples of 2,3 and 5 from 1 to 2x3x5=30
You are then left with the set {1, 7, 11, 13, 17, 19, 23, 29}
For P=7, you would remove all the multiples of 2,3,5,7 from 1 to 2x3x5x7=210
You are then left with the set {1, 11, 13, 17, ......... 199, 209} etc.
(Just for the record, I am aware this does NOT generate a set of prime numbers, e.g. 209 = 11x19)
My question is, for a certain P, what is the maximum gap / difference between successive elements of the set and how do I go about proving it?
When P=3 with the set {1, 5}, the maximum gap is 4
When P=5 with the set {1, 7, 11, 13, 17, 19, 23, 29}, the maximum gap is 6
When P=7 with the set {1, 11, 13, 17, .... 199, 209}, the maximum gap is 10
When P=11, the maximum gap is 14
I am currently putting together a program to calculate maximum gaps for higher values of P.
Once I am confident of a pattern, how could I go about proving this (maximum gaps)? I have no idea where to even begin!
Thanks.