A234960 Sequence (or tree) T of primes generated by these rules: 2 is in T; if p is in T, then the greatest prime < 2*p is in T; if p is in T, then the least prime > 2*p is in T; duplicates are deleted as they occur.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 73, 79, 83, 89, 97, 103, 107, 113, 127, 131, 137, 139, 149, 157, 163, 167, 173, 179, 193, 197, 199, 211, 223, 227, 251, 257, 263, 271, 277, 281, 293, 307, 313, 317, 331, 337, 347, 353

Offset

1,1

Comments

The rules generate successive generations g(n) as follows: g(1) = (2), which begets 3 and 5, so that g(2) = (3,5); then g(3) = (7,11); g(4) = (13,17,19,23); etc. The number of primes in g(n) is given by A234961, and primes not generated, beginning with 71, are given by A234962. Conjecture: the limiting relative density of generated primes is 0.

Links

Clark Kimberling, Table of n, a(n) for n = 1..4000

Example

Starting with 2, the greatest prime less than 2*2 is 3, and the least prime greater than 2*2 is 5.

Mathematica

t = NestList[DeleteDuplicates[Flatten[Map[{#, NextPrime[2 #, -1], NextPrime[2 #, 1]} &, #]]] &, {2}, 9]; g = Join[{{2}}, Map[Complement[t[[# + 1]], t[[#]]] &, Range[Length[t] - 1]]]
Flatten[g] (* A234960 *) (* Peter J. C. Moses, Dec 30 2013 *)

Prog

(Python)
from sympy import prevprime, nextprime
def aupto(limit):
    reach, expand = {2}, [2]
    while True:
        newreach = set()
        while len(expand) > 0:
            p = expand.pop()
            for q in prevprime(2*p), nextprime(2*p):
                if q not in reach:
                    newreach.add(q)
        reach |= newreach
        expand = list(newreach)
        if prevprime(2*min(expand)) > limit:
            return sorted(r for r in reach if r <= limit)
print(aupto(353)) # Michael S. Branicky, Jul 24 2022

Crossrefs

Cf. A234961, A234962.

Sequence in context: A176165 A196230 A233360 * A118850 A322443 A219697

Adjacent sequences: A234957 A234958 A234959 * A234961 A234962 A234963

Keyword

nonn,easy

Author

Clark Kimberling, Jan 01 2014

Status

approved