Are there Generalizations of this theorem I found on a chit of paper?

Question

Some time ago , I was reading a book which was not of a mathematical taste from a library . But from that book a chit of paper came out which was handwritten and had the title : Aubry's theorem And it read

                              Aubry's theorem 

Any group of 30 consecutive odd numbers does not contain more than 15 prime numbers or numbers all of whose prime factors exceed 7.

It was a very curious result for me and I turned the paper back and forth but what I could find was just a date :$25 -3-97$ and nothing else.

My question here is are there generalizations of this result , What changes can we make in those pairs of numbers $(30,15,7)$ ?

Answer 1

This is the basis of the number wheel in prime sieve algorithms, in this case the 7-wheel.

The step before is the 3-wheel, which gives that any prime number must have the form $6k\pm1$. Using additionally the factor 5 gives the 5-wheel with a period of 30, and among the remaining remainders

 1, 5, 7, 11, 13, 17, 19, 23, 25, 29

the ones divisible by 5 will never belong to prime numbers, reducing the list to

 1, 7, 11, 13, 17, 19, 23, 29

resulting in 8 prime number candidates among every segment of 15 odd numbers.

One can further refine this argument by using the 7-wheel, which has a period of 210 and excludes remainders divisible by 7

  1,   _   11,  13,  17,  19,  23,  29, 
 31,  37,  41,  43,  47,   _   53,  59, 
 61,  67,  71,  73,   _   79,  83,  89, 
  _   97, 101, 103, 107, 109, 113, 119,
121, 127, 131,  _   137, 139, 143, 149,
151, 157,  _   163, 167, 169, 173, 179,
181, 187, 191, 193, 197, 199,  _ , 209

and indeed in every segment of 60 numbers or 30 odd numbers there is at least one remainder eliminated as divisible by 7.