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A371694
a(1) = 2; a(n+1) is the larger prime between nextprime(a(n)) and prevprime(a(n)+n-m+1), where m is the number of primes < a(n) that are missing from the sequence.
0
2, 3, 5, 7, 11, 13, 19, 23, 29, 37, 43, 47, 53, 61, 71, 79, 89, 97, 107, 113, 127, 137, 139, 151, 163, 173, 181, 193, 199, 211, 223, 233, 241, 251, 263, 277, 283, 293, 307, 317, 331, 337, 353, 367, 379, 389, 401, 409, 421, 433, 449, 463, 467, 479, 491, 503
OFFSET
1,1
COMMENTS
The first missing prime in the sequence is 17. First occurrence of a(n)+n-m < nextprime(a(n)) is at n = 20 (see Examples). It seems that 1/2 < n/(n+m) <= 1 and lim_{n->oo} n/(n+m) = 1/2 (or half of the primes are in this sequence).
EXAMPLE
primes 2 3 5 7 11 13 17 19 23 29 31 37 41 43 .. 97 101 103 107 109 113 127
n 1 2 3 4 5 6 7 8 9 10 11 .. 18 19 20 21
a(n) 2 3 5 7 11 13 19 23 29 37 43 .. 97 107 113 127
m 0 0 0 0 0 0 1 1 1 2 3 .. 7 9 10 10
a(n)+n-m 3 5 8 11 16 19 25 30 37 45 51 .. 108 117 123 138
a(n+1) 3 5 7 11 13 19 23 29 37 43 47 .. 107 113 127 137
PROG
(Python)
from sympy import primerange, prevprime, nextprime; p = 2; b = 0
for n in range(1, 57): print(p, end = ", "); q = max(nextprime(p), prevprime(p + n - b + 1)); m = len(list(primerange(p+1, q))); p = q; b += m
CROSSREFS
Cf. A362527.
Sequence in context: A005728 A050437 A096246 * A106639 A233462 A233893
KEYWORD
nonn
AUTHOR
Ya-Ping Lu, Apr 03 2024
STATUS
approved