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A275246 Sequence of pairwise relatively prime numbers of class P_3 (see comment). 9
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 (list; graph; refs; listen; history; text; internal format)
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 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).

LINKS

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

MATHEMATICA

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

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

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Last modified April 18 22:08 EDT 2019. Contains 322237 sequences. (Running on oeis4.)