A275246 Sequence of pairwise relatively prime numbers of class P_3 (see comment).

8, 15, 77, 221, 437, 899, 1517, 2021, 3127, 4087, 5183, 6557, 8633, 10403, 11663, 14351, 17947, 20711, 23707, 27221, 30967, 34571, 38021, 41989, 50621, 53357, 57599, 64507, 70747, 75067, 79523, 89951, 97343, 104927, 116939, 123197, 131753, 141367, 148987

Offset

1,1

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_(n-1)and to obtain p_(n+1) we should remove all remaining 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 algorithm 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).

Links

Table of n, a(n) for n=1..39.

Formula

a(n) = prime(2*n-2)*prime(2*n-1), n>1, a(1)=8. - Bill McEachen, Nov 26 2024

Mathematica

k = 3; {2 (k + 1)}~Join~Map[Times @@ # &, Partition[Prime@ Range[k - 1, 78], 2, 2]] (* Michael De Vlieger, Jul 21 2016 *)

Prog

(PARI) a(n)=if(n==1, 8,  prime(2*n-2)*prime(2*n-1)) \\ Bill McEachen, Nov 26 2024

Crossrefs

Cf. A000040, A001248, A089581, A275248, A275249, A275251, A275252, A275253.

Sequence in context: A193490 A015442 A253211 * A177199 A177165 A189003

Adjacent sequences: A275243 A275244 A275245 * A275247 A275248 A275249

Keyword

nonn

Author

Vladimir Shevelev and Peter J. C. Moses, Jul 21 2016

Status

approved