A355273 Primes p for which p + q is a multiple of 4, where q is the previous prime if p == 2 (mod 3) or the next prime otherwise.

3, 5, 29, 31, 53, 59, 61, 73, 89, 137, 139, 149, 151, 157, 173, 179, 181, 191, 239, 241, 251, 257, 263, 269, 271, 283, 293, 331, 337, 347, 359, 367, 373, 389, 409, 419, 421, 431, 433, 449, 509, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 631

Offset: 1

Author

M. F. Hasler and Yasutoshi Kohmoto, Jun 26 2022

Keywords

• nonn

Comments

Naively one might expect p + precprime / nextprime congruent to 0 or to 2 (mod 4) with equal probability. It turns out that, following the given rule, the case 2 is much more frequent than the case 0, especially for small primes. (Observation by Y. Kohmoto.)

See the comment from 2017 in A068228 for an explanation.

Crossrefs

Cf. A151799 (previous prime), A151800 (next prime).

Cf. A068228.

Programs

PARI

select( is(p)=if(p%3==2, precprime(p-1)+p, nextprime(p+1)+p)%4==0, primes(149))

Python

from sympy import nextprime
from itertools import islice
def agen():
    p, q = 2, [3, 1]
    while True:
        if (p + q[int(p%3 == 2)])%4 == 0: yield p
        p, q = q[0], [nextprime(q[0]), p]
print(list(islice(agen(), 54))) # Michael S. Branicky, Jun 26 2022