Groups of primes in a limited range

In the range until 1,000,000

The ratio of the number of P+ to the number of P- in this range is equal p+/ p-= 1.00015 ≈1.

Unlike in the infinite general set, 
every ratio of a number of groups of next bigger size to that of groups of previous size is less than 2, 
and it increases when the sizes of groups increase.

The only 11-group contains P+.

         
  Ord# Pmod6  
  34368 406907 5  
  34369 406951 1  
  34370 406969 1  
  34371 406981 1  
  34372 406993 1  
  34373 407023 1  
  34374 407047 1  
  34375 407059 1  
  34376 407083 1  
  34377 407119 1  
  34378 407137 1  
  34379 407149 1  
  34380 407153 5  
         

 

Thus,       ~1/3 primes of a class go single (more than in the infinite general set),
~1/3 ones go in pairs (more than in the infinite general set),
             and ~1/3 ones go in groups of size > 2 (less than in the infinite general set).

~1/6 go inside groups of more than 2 consecutive Pk (have V0);                                  

~1/2 of Pk appear as the first or last members of  nonsingle groups of consecutivePk, 

          and ~1/2 of them have an ajacent single PR (found by sorting primes by class).

N(V0) 1/6 p(x) c1

Additionally, it is possible to consider the portions of V + and V- are equal, so each of them is 5/12, and the ratio of such a portion to the portion of V 0 is equal approximately (5/12)/(1/6) = 5/2 = 2.5. But, the ratio M/m for regressions of distribution of absolute values of variations in the range until 200,000,000 is equal 2.498 ≈ 2.5, too!

Let's research this analytically and check numerically

Because only last members of groups have the next prime of an opposite class


For this range 

 The reason of different condition in a finite set of primes is hidden in real primes' distribution in groups of the same class. 
Singles create next primes of an opposite class only (both gaps and variations, not multiple of 6 only).
 Pairs do equal numbers of next primes of both the classes (equal numbers of gaps multiple of 6 and not; and variations not multiple of 6, only) . 
And groups of n > 2  create more next primes of the same class than ones of an opposite class (more gaps , multiple of 6, than not; and variations multiple of 6 for all inner primes of groups). 
The predominance of the role of singles over the role of primes in groups of sizes more than 2 (maximally trice) is a reason of being p(Pk) < in a finite set of primes.

666666666666

Maximal groups of primes of the same class

In the range until 10,000,000

The only 13-group contains P+.

In the range until 35,000,000

The only 15-group contains P-.

In the range until 100,000,000

Two 16-groups contain P+.

 

Up ] Introduction of Variation ] Regular and Repeating Groups ] Groups in the Range until 100,000 ] Groups in the Range until 1,000,000 ] Key of Primes Structure ] Distribution of gaps and variations ] Boolean Algebra of Classes ] Consecutive primes ] Groups of primes in the infinite set ] [ Groups of primes in a limited range ] We have for each kind of regular and repeating groups ]