A341224 - OEIS

%I #19 Feb 07 2021 14:02:08

%S 3,163,181,191,229,251,839,971,1181,1201,1489,1801,1823,1847,1861,

%T 1987,2069,2087,3547,3691,3697,6361,6637,6899,6967,7793,7963,8731,

%U 8737,10253,10271,10613,10639,10799,11981,12689,12697,13697,13841,14951,15299,16547,16747

%N Primes p such that (p^32 + 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, and this sequence list primes of the form (p^j + 1)/2 for j=2^0=1, j=2^1=2, ..., j=2^5=32, respectively.

%H Jon E. Schoenfield, <a href="/A341224/b341224.txt">Table of n, a(n) for n = 1..10000</a>

%e (3^32 + 1)/2 = 926510094425921 is prime, so 3 is a term.

%e (5^32 + 1)/2 = 11641532182693481445313 = 641*75068993*241931001601, so 5 is not a term.

%p q:= p-> (q-> q(p) and q((p^32+1)/2))(isprime):

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

%t Select[Range[17000], PrimeQ[#] && PrimeQ[(#^32 + 1)/2] &] (* _Amiram Eldar_, Feb 07 2021 *)

%o (PARI) isok(p) = (p>2) && isprime(p) && isprime((p^32 + 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), (this sequence) (k=5).

%Y Cf. A000040.

%K nonn

%O 1,1

%A _Jon E. Schoenfield_, Feb 07 2021