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A340480 Primes p such that (p^8 + 1)/2 is prime. 9
13, 43, 47, 53, 239, 373, 409, 433, 491, 557, 577, 859, 1021, 1103, 1307, 1531, 1699, 1753, 1777, 1871, 2053, 2083, 2297, 2467, 2503, 2593, 2797, 2957, 3251, 3307, 3323, 3511, 3613, 4099, 4523, 4637, 4951, 4999, 5591, 5657, 5693, 5801, 5827, 5849, 6043, 6163 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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, and this sequence list primes of the form (p^j + 1)/2 for j=2^0=1, j=2^1=2, j=2^2=4, and j=2^3=8, respectively.
(p^8 + 1)/2 is divisible by 17 when m mod 34 is 3, 5, 7, 11, 23, 27, 29, or 31.
LINKS
EXAMPLE
(3^8 + 1)/2 = 3281 = 17*193, so 3 is not a term.
(13^8 + 1)/2 = 407865361 is prime, so 13 is a term.
(17^8 + 1)/2 = 3487878721 = 18913 * 184417, so 17 is not a term.
MATHEMATICA
Prime[Position[Table[(Prime[p]^8 + 1)/2, {p, 1, 803}], _Integer?PrimeQ]] // Flatten (* Robert P. P. McKone, Jan 31 2021 *)
PROG
(PARI) isok(p) = isprime(p) && (p>2) && isprime((p^8 + 1)/2); \\ Michel Marcus, Feb 01 2021
CROSSREFS
Primes p such that (p^(2^k) + 1)/2 is prime: A005383 (k=0), A048161 (k=1), A176116 (k=2), (this sequence) (k=3).
Sequence in context: A259427 A118857 A118529 * A129811 A108545 A136191
KEYWORD
nonn
AUTHOR
Jon E. Schoenfield, Jan 31 2021
STATUS
approved

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Last modified May 22 20:45 EDT 2024. Contains 372758 sequences. (Running on oeis4.)