Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #41 Nov 27 2024 23:02:45
%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).
%F a(n) = prime(2*n-2)*prime(2*n-1), n>1, a(1)=8. - _Bill McEachen_, Nov 26 2024
%t k = 3; {2 (k + 1)}~Join~Map[Times @@ # &, Partition[Prime@ Range[k - 1, 78], 2, 2]] (* _Michael De Vlieger_, Jul 21 2016 *)
%o (PARI) a(n)=if(n==1, 8, prime(2*n-2)*prime(2*n-1)) \\ _Bill McEachen_, Nov 26 2024
%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