%I #15 Feb 07 2021 23:30:52

%S 3,353,587,727,863,883,919,1217,1237,1657,2029,2203,2333,3209,3529,

%T 3617,3889,4889,5387,5557,5689,5749,6701,6961,7727,8443,9377,9433,

%U 10009,10243,10691,10799,11027,12071,12451,13681,13687,15569,15601,15823,16759,17939

%N Primes p such that (p^64 + 1)/2 is prime.

%C Expressions of the form m^j + 1 can be factored (e.g., m^3 + 1 = (m + 1)*(m^2 - m + 1)) for any positive integer j except when j is a power of 2, so (p^j + 1)/2 for prime p cannot be prime unless j is a power of 2. A005383, A048161, A176116, A340480, A341210, A341224, and this sequence list primes of the form (p^j + 1)/2 for j=2^0=1, j=2^1=2, ..., j=2^6=64, respectively.

%H Jon E. Schoenfield, <a href="/A341229/b341229.txt">Table of n, a(n) for n = 1..10000</a>

%e (3^64 + 1)/2 = 1716841910146256242328924544641 is prime, so 3 is a term.

%e (5^64 + 1)/2 = 271050543121376108501863200217485427856445313 = 769*3666499598977*96132956782643741951225664001, so 5 is not a term.

%p q:= p-> (q-> q(p) and q((p^64+1)/2))(isprime):

%p select(q, [$3..20000])[]; # _Alois P. Heinz_, Feb 07 2021

%t Select[Range[18000], PrimeQ[#] && PrimeQ[(#^64 + 1)/2] &] (* _Amiram Eldar_, Feb 07 2021 *)

%o (PARI) isok(p) = (p>2) && isprime(p) && ispseudoprime((p^64 + 1)/2); \\ _Michel Marcus_, Feb 07 2021

%Y Primes p such that (p^(2^k) + 1)/2 is prime: A005383 (k=0), A048161 (k=1), A176116 (k=2), A340480 (k=3), A341210 (k=4), A341224 (k=5), (this sequence) (k=6).

%K nonn

%O 1,1

%A _Jon E. Schoenfield_, Feb 07 2021