%I #43 Dec 05 2023 18:52:37

%S 2,3,4,5,7,8,16,19,29,47,59,163,257,421,937,947,1493,1901,6689,8087,

%T 9679,28753,79043,129127,145969,165799,168677,170413,172243,278321,

%U 552283

%N Indices of prime companion Pell numbers, divided by 2 (A001333).

%C Note that for A001333(n) to be prime, the index n must be prime or a power of 2. The indices greater than 421 yield probable primes.

%C Numbers n for which ((1+sqrt(2))^n + (1-sqrt(2))^n)/2 is prime. - _Artur Jasinski_, Dec 10 2006

%D F. Le Lionnais, Les Nombres Remarquables. Paris: Hermann, p. 62, 1983.

%H J. B. Cosgrave and K. Dilcher, <a href="https://fq.math.ca/Papers1/51-2/CosgraveDilcher-1.pdf">Pairs of reciprocal quadratic congruences involving primes</a>, Fib. Quart. 51 (2) (2013) 98, after Theorem 3.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PellNumber.html">Pell Number</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IntegerSequencePrimes.html">Integer Sequence Primes </a>

%t lst={}; a=1; b=1; Do[c=a+2b; a=b; b=c; If[PrimeQ[c], AppendTo[lst, n]], {n, 2, 10000}]; lst

%t (* Second program: *)

%t Do[If[PrimeQ[Expand[((1 + Sqrt[2])^n + (1 - Sqrt[2])^n)/2]], Print[n]], {n, 0, 1000}] (* _Artur Jasinski_, Dec 10 2006 *)

%o (PARI) isok(n) = isprime(polchebyshev(n, 1, I)/I^n); \\ _Michel Marcus_, Dec 07 2018

%Y Cf. A002203 (companion Pell numbers), A086395 (primes in A001333), A096650 (indices of prime Pell numbers).

%Y Cf. A005850.

%K hard,nonn

%O 1,1

%A _T. D. Noe_, Sep 24 2004

%E a(24) from _Eric W. Weisstein_, May 22 2006

%E a(25) from _Eric W. Weisstein_, Aug 29 2006

%E a(26) from _Eric W. Weisstein_, Nov 11 2006

%E a(27) from _Eric W. Weisstein_, Nov 26 2006

%E a(28) from _Eric W. Weisstein_, Dec 10 2006

%E a(29) from _Eric W. Weisstein_, Jan 25 2007

%E a(30) from _Robert Price_, Dec 07 2018

%E a(31) from _Robert Price_, Dec 05 2023