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A382669 Even numbers m such that both p = m^2 + 1 and q = (p^2 + 1)/2 are primes.

%I A382669 #35 May 02 2025 11:29:44
%S A382669 2,10,150,160,230,270,400,890,910,920,1060,1430,1550,1970,2700,2960,
%T A382669 3280,3290,3520,3660,4140,4330,4510,4700,4780,4850,4920,5180,5360,
%U A382669 5500,5560,5620,5880,5960,6220,6460,6980,7160,7190,7520,7550,7820,9630,9760,9900
%N A382669 Even numbers m such that both p = m^2 + 1 and q = (p^2 + 1)/2 are primes.
%C A382669 Except 2, all terms are divisible by 10, and p-1 and q-1 are divisible by 100.
%C A382669 Numbers m such that p = m^2+1 and p + m^4/2 are both prime. - _Chai Wah Wu_, May 01 2025
%H A382669 Robert Israel, <a href="/A382669/b382669.txt">Table of n, a(n) for n = 1..10000</a>
%e A382669 10 is a term because both 10^2 + 1 = 101 and (101^2 + 1)/2 = 5101 are primes.
%p A382669 filter:= proc(m) local p;
%p A382669   p:= m^2 + 1;
%p A382669   isprime(p) and isprime((p^2+1)/2)
%p A382669 end proc:
%p A382669 select(filter, [2,seq(i,i=10..10000,10)]); # _Robert Israel_, May 02 2025
%t A382669 Select[2*Range[5000], PrimeQ[#^2 + 1] && PrimeQ[#^4/2 + #^2 + 1] &] (* _Amiram Eldar_, Apr 24 2025 *)
%o A382669 (Python)
%o A382669 from sympy import isprime
%o A382669 for n in range(2, 10000, 2): x = n*n + 1; ct = 0; print(n, end = ', ') if isprime(x) and isprime((x*x + 1)//2) else 0
%o A382669 (Python)
%o A382669 from itertools import count, islice
%o A382669 from sympy import isprime
%o A382669 def A382669_gen(): # generator of terms
%o A382669     yield 2
%o A382669     yield from filter(lambda m: isprime(p:=m**2+1) and isprime(p+(m**4>>1)),(10*k for k in count(1)))
%o A382669 A382669_list = list(islice(A382669_gen(),45)) # _Chai Wah Wu_, May 02 2025
%Y A382669 Cf. A002522, A005574, A048161.
%K A382669 nonn
%O A382669 1,1
%A A382669 _Ya-Ping Lu_, Apr 24 2025