
COMMENTS

Consider the sequence P_0 of primes (A000040).
The simplest algorithm giving this sequence is the sieve of Eratosthenes. If we already know primes 2,3,...,p_n, then, by the algorithm of this sieve, the remaining numbers are not divisible by 2,3,...,p_(n1)and to obtain p_(n+1) we should remove all remaing numbers divisible by p_n.
Note that we can also say that we remove all remaining numbers k for which GCD(k,p_n)>1. Although for generating the primes the algorithm is unchanged, in this form the algorithm we will apply in more general cases. Denote this agorithm by E*.
Remove 1 and the primes from the positive numbers. We get sequence
4,6,8,9,10,12,14,15,16,18,20,21,22,24,... (1)
By algorithm E*, keeping 4, we remove all even numbers; further keeping 4,9, we remove numbers divisible by 3, etc. We obtain sequence 4,9,25,49,...consisting of squares of primes (A001248). Denote this sequence by P_1. Removing P_1 from (1), we obtain sequence
6,8,10,12,14,15,16,18,20,21,22,24,26,... (2)
By algorithm E*, keeping 6, we remove all numbers divisible by 2 and 3; the least ramaning number is 35; keeping 6 and 35, we remove further all numbers divisible by 5 and 7, etc. We obtain sequence 6,35,143,...
(A089581). Denote this sequence by P_2.
The sequence P_3,...,P_8 are presented in A275246, A275248, A275249, A275251, A275252, A275253 respectively. All sequences {P_k} consist of pairwise relatively prime numbers, beginning with 2*(k+1)(which is a unique even number in sequence P_k).
