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
Keywords
Examples
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.
Links
- Robert Israel, Table of n, a(n) for n = 1..2500
Programs
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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 *) -
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
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