How useful can it be to study how prime number sequences interact?


Starting from the number 5, the sequence obtainable by joining the sequences of numbers 2 and 3 (which I call (Sc<=3)) is obtained here by adding to the previous number (regardless of whether the resulting number is discarded from the sequence of other prime numbers) the numbers 2 and 4 in succession.

The intervention of the next sequence (in this case of the number 5) shows me the next sequence of numbers that added together create the sequence also composed of the number 5.

+2=7

+4=11

+2=13

+4=17

+2=19

+4=23

+6=29 (6 <– +2=25 +4)

+2=31

+6=37 (6 <– +4=35 +2)

+4=41

+2=43

+4=47

+6=53 (6 <– +2=49 +4)

+6=59 (6 <– +2=55 +4)

+2=61

+6=67 (6 <– +4=65 +2)

+4=71

+2=73

+6=79 (6 <– +4=77 +2)

+4=83

+6=89 (6 <– +2=85 +4)

+8=97 (8 <– +2=91 +4=95 +2)

+4=101

+2=103

+4=107

+2=109

+4=113

+14=127 (14 <– +2=115 +4=119 +2=121 +4=125 +2)

Starting from the number 7, if we want to obtain the same result using the sequence obtainable by joining the sequences of the numbers 2, 3 and 5 (which I call (Sc<=5)) we must add the numbers 4, 2, 4, 2, 4, 6, 2, 6 in succession.

+4=11 +2=13 +4=17 +2=19 +4=23 +6=29 +2=31 +6=37 …

I note 4+2+4+2+4+6+2+6 = 2 * 3 * 5

Likewise, I think it is possible to continue with the next sequences.

I see (Sc<=3) as a good starting point, on which (starting from their square) the successive prime numbers interact.
 
Everything would be simpler if the subsequent primes did not find some of the work already done by the previous primes.

However, every single sequence of prime numbers is regular and improves the definition of the overall distribution of prime numbers.