1,1

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, A341229, A341230, A341234, A341264, and this sequence list primes of the form (p^j + 1)/2 for j=2^0=1, j=2^1=2, ..., j=2^10=1024, respectively.

(3^1024 + 1)/2 = 1866959243...6855178241 (a 489-digit number) = 59393 * 448524289 * 847036417 * 8273923970...2296603649 (a 466-digit composite number), so 3 is not a term.

(827^1024 + 1)/2 = 1677304013...0116613121 (a 2988-digit number) is prime, so 827 is a term. Since 827 is the smallest prime p such that (p^1024 + 1)/2 is prime, it is a(1) and is also A341211(10).

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), A341229 (k=6), A341230 (k=7), A341234 (k=8), A341264 (k=9), (this sequence) (k=10).

Cf. A341211 (Smallest prime p such that (p^(2^n) + 1)/2 is prime).

Sequence in context: A046496 A102350 A108830 * A340115 A134726 A104281

Adjacent sequences: A341269 A341270 A341271 * A341273 A341274 A341275

Jon E. Schoenfield, Feb 07 2021

a(17)-a(35) from Jinyuan Wang, Feb 09 2021

approved