Some patterns of sets and subsets of prime divisors of natural numbers of a certain type.

When I studied mathematics, I became interested in prime numbers and formulas that give a large number of prime numbers. Having become acquainted with the Fermat and Mersen formulas, which are fast growing and the Euler formula, which is slow growing, I tried to come up with a formula that is intermediate between the fast growing and slow growing formulas that give a large number of prime numbers. The most interesting was the following formula: X=(C^(2^n ) 1)/2 (1.) , where C is any natural odd number greater than one (the set C can be extended to all integers except zero, but I considered this to be redundant for solving the tasks set for myself), and n is any natural number. The set of values of X for any (within the above limits) values of C and n contains quite a large number of prime numbers. As C and especially n increase, the number of prime values of the set X decreases as expected. An infinite set X can be divided into subsets X1, X2, X3, …. and so on in accordance with the degree of n in the formula (1.), from 1 to infinity in the series of natural numbers. Associated with the set X is the set P which includes the set of all prime divisors of the numbers included in the set X. In turn, the set P can be divided into an infinite number of subsets, by the value of the degree n - from P1, P2, P3, ……… and further to infinity in the series of natural numbers. Each subset of the set of values X of a certain degree n (Xn) corresponds to a subset of the set P of the same degree n (Pn), since it contains all the prime divisors of the members of this subset. Among the above subsets of the set P, only the subset P1 (with exponent n=1 of formula (1.)), has two features in comparison with other subsets (with exponent n more than 1): 1. Only when n=1 (formula (1.)) and in cases where odd C, in the decimal number system, is a number ending in digits 3 or 7, then X always takes a value that is a multiple of 5, or numbers that are natural powers of a number 5. Since the given prime divisor of the numbers X never occurs in the subsets Pn of the set P for n more than 1, I do not include the number 5 in the set P; 2. Only when n=1 (formula (1.)) among the divisors of numbers X there are simple divisors in a degree greater than 1 (in 2 and 4 powers). For n more than 1, among the divisors of the numbers X, there are prime divisors only to the power of 1. This feature, in my opinion, is important and interesting. Next, let’s move on to the patterns that have been identified in all subsets of the sets X and P: 1. In each subset of the set P, which includes all prime divisors of the members of the corresponding subset of the set X, there cannot be prime divisors less than a certain exponent p. The value of this smaller divisor p increases with the growth of the value of the degree n (formula (1.)) in a rather complicated way. Namely: For n=1, among the divisors of the subset X1 and, accordingly, the members of the subset P1, there are no prime divisors less than 13. The prime divisor 13 is therefore the value of p1. For n=2 and n=3, among the prime divisors of the subsets X2 and X3, and accordingly the members of the subsets P2 and P3, there are no prime divisors less than 17. The prime divisor 17 is therefore the value of the exponents p2 and p3. The value of all the p indicators I have identified are given in the following table: Table 1. p1 13 P2 17 P3 17 P4 97 P5 193 P6 257 P7 257 P8 12289 P9 134382593 I did not calculate subsequent (greater than p9) indicators p due to my lack of necessary tools for their calculation, since at n = 10 (formula (1.)) the number of digits of the minimum value of the number X exceeds 600. Based on the data in Table 1, it can be assumed that the values of the indicators (minimum divisors) p10, p11, as well as p14, p15, and so on ad infinitum, will be the same in each such pair. Even bolder assumptions might be: - the numerical values of all the above consecutive pairs with the same values will end with the number 7 (hereinafter in decimal notation); - numeric values p1, p5, p9, p13, p17 and further to infinity, will end with the number 3, or even the number 93; - the number of repeated digits in the last two assumptions above can lengthen to the left with each step to infinity. 2. The set P (prime divisors of the set X) (formula (1.)) does not include all prime numbers, but only some of them. With an increase in the degree n (formula (1.)) the distance between prime divisors of subsets Xn in subsets Pn mainly increases, due to the disappearance of a significant part of prime divisors from subsets Pn with smaller n in subsets Pn with larger n. Although a significant coincidence of prime divisors in subsets of Pn with an increase in the degree of n between them remains. Also, with an increase in the degree of n (for