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.

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

Author

Ya-Ping Lu, Apr 03 2024

Keywords

• nonn

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).

Examples

			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
		

Crossrefs

Cf. A362527.

Programs

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