|
%I #30 Aug 28 2023 08:22:47
%S 8,15,77,221,437,899,1517,2021,3127,4087,5183,6557,8633,10403,11663,
%T 14351,17947,20711,23707,27221,30967,34571,38021,41989,50621,53357,
%U 57599,64507,70747,75067,79523,89951,97343,104927,116939,123197,131753,141367,148987
%N Sequence of pairwise relatively prime numbers of class P_3 (see comment).
%C Consider the sequence P_0 of primes (A000040).
%C 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_(n-1)and to obtain p_(n+1) we should remove all remaining numbers divisible by p_n.
%C 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 algorithm by E*.
%C Remove 1 and the primes from the positive numbers. We get sequence
%C 4,6,8,9,10,12,14,15,16,18,20,21,22,24,... (1)
%C 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
%C 6,8,10,12,14,15,16,18,20,21,22,24,26,... (2)
%C 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,...
%C (A089581). Denote this sequence by P_2.
%C 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).
%t k = 3; {2 (k + 1)}~Join~Map[Times @@ # &, Partition[Prime@ Range[k - 1, 78], 2, 2]] (* _Michael De Vlieger_, Jul 21 2016 *)
%Y Cf. A000040, A001248, A089581, A275248, A275249, A275251, A275252, A275253.
%K nonn
%O 1,1
%A _Vladimir Shevelev_ and _Peter J. C. Moses_, Jul 21 2016
|
