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URL: https://oeis.org/A245674

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A245674
Prime numbers P such that 8*P^2-1 and 8*(8*P^2-1)^2-1 are also prime numbers.
1
2, 79, 107, 173, 257, 359, 383, 523, 593, 971, 1493, 1811, 1867, 2273, 2357, 3187, 4111, 4723, 6389, 7607, 8101, 8699, 9473, 11027, 12157, 12227, 15017, 16301, 16987, 18797, 19801, 19913, 20071, 20323, 21313, 22003, 22307, 23203, 24229, 24733, 24859, 24943
OFFSET
1,1
COMMENTS
Subsequence of A245639.
For P < 10^9 in this sequence, 8*(8*(8*P^2-1)^2-1)^2-1 is composite.
Let f(x) = 8*x^2-1 and P >= 2. Then {f(P), f(f(P)), f(f(f(P)))} cannot all be prime. Proof: By doing cases on P mod 7, it can be shown that {f(P), f(f(P)), f(f(f(P)))} contains a multiple of 7. Also, all 3 numbers are greater than 7. - Jason Yuen, Feb 26 2025
EXAMPLE
2 is prime, 8*2^2-1=31 is prime, 8*31^2-1=7687 is prime, so 2 is in the sequence.
MATHEMATICA
f[n_]:=8 n^2 - 1; Select[Prime[Range[3000]], PrimeQ[f[#]]&&PrimeQ[f[f[#]]]&] (* Vincenzo Librandi, Sep 08 2014 *)
PROG
(PFGW)
SCRIPT
DIM i
DIM j, 0
DIM n, 0
DIMS t
OPENFILEOUT myf, a(n).txt
LABEL loop1
SET j, j+1
SETS t, %d\ ; p(j)
SET i, 8*p(j)^2-1
PRP i, t
IF ISPRP THEN GOTO a
GOTO loop1
LABEL a
SET i, 8*i^2-1
PRP i, t
IF ISPRP THEN GOTO b
GOTO loop1
LABEL b
WRITE myf, t
GOTO loop1
(PARI)
f(x) = 8*x^2-1;
forprime(p=1, 10^5, if(ispseudoprime(f(p)) && ispseudoprime(f(f(p))), print1(p, ", "))) \\ Derek Orr, Jul 29 2014
(Magma) [p: p in PrimesUpTo(25000)| IsPrime(8*p^2-1)and IsPrime(512*p^4-128*p^2+7)]; // Vincenzo Librandi, Sep 08 2014
CROSSREFS
Cf. A245639.
Sequence in context: A231240 A392485 A197101 * A342979 A166052 A045484
KEYWORD
nonn,easy
AUTHOR
Pierre CAMI, Jul 29 2014
STATUS
approved