A352951 Primes p such that p+2, (p^2-5)/2-p, (p^2-1)/2+p, and (p^2+3)/2+3*p are all prime.

5, 29, 599, 26699, 59669, 72869, 189389, 285839, 389999, 508619, 623669, 708989, 862229, 908879, 945629, 945809, 953789, 1002149, 1134389, 1138409, 1431569, 1461209, 1712549, 2110289, 2127269, 2158589, 2704769, 2727299, 2837279, 3004049, 3068909, 3091379, 3280229, 3336659, 3402239, 3546269

Offset

1,1

Comments

Lower twin primes p such that if q = p+2, then (p*q-1)/2, (p*q-1)/2-p-q and (p*q-1)/2+p+q are also prime.

All terms but the first == 29 (mod 30).

Links

Robert Israel, Table of n, a(n) for n = 1..2500

Example

a(3)=599 is a term because it, 599+2 = 601, (599*601-1)/2 = 179999, 179999-599-601 = 178799, and 179999+599+601 = 181199 are prime.

Maple

R:= 5: count:= 0:
for p from 29 by 30 while count < 60 do
  if isprime(p) and isprime(p+2) then
    q:= p+2; r:= (p*q-1)/2;
    if isprime(r) and isprime(r+p+q) and isprime(r-p-q) then
      count:= count+1; R:= R, p;
    fi
  fi
od:
R;

Mathematica

Select[Prime[Range[250000]], And @@ PrimeQ[{# + 2, (#^2 - 5)/2 - #, (#^2 - 1)/2 + #, (#^2 + 3)/2 + 3*#}] &] (* Amiram Eldar, Apr 11 2022 *)
Select[Prime[Range[260000]], AllTrue[{#+2, (#^2-5)/2-#, (#^2-1)/2+#, (#^2+3)/2+3#}, PrimeQ]&] (* Harvey P. Dale, Jun 12 2024 *)

Prog

(Python)
from itertools import islice
from sympy import isprime, nextprime
def agen(): # generator of terms
    p, q = 3, 5
    while True:
        if q == p+2:
            t, s = (p*q-1)//2, p+q
            if isprime(t) and isprime(t+s) and isprime(t-s):
                yield p
        p, q = q, nextprime(q)
print(list(islice(agen(), 36))) # Michael S. Branicky, Apr 10 2022

Crossrefs

Cf. A352948.

Subsequence of A001359.

Sequence in context: A259534 A176680 A352948 * A069142 A144994 A263369

Adjacent sequences: A352948 A352949 A352950 * A352952 A352953 A352954

Keyword

nonn

Author

J. M. Bergot and Robert Israel, Apr 10 2022

Status

approved