A099088 Indices of prime companion Pell numbers, divided by 2 (A001333).

2, 3, 4, 5, 7, 8, 16, 19, 29, 47, 59, 163, 257, 421, 937, 947, 1493, 1901, 6689, 8087, 9679, 28753, 79043, 129127, 145969, 165799, 168677, 170413, 172243, 278321, 552283

Offset: 1

Author

T. D. Noe, Sep 24 2004

Keywords

• hard
• nonn

Comments

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.

Numbers n for which ((1+sqrt(2))^n + (1-sqrt(2))^n)/2 is prime. - Artur Jasinski, Dec 10 2006

References

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

Links

• J. B. Cosgrave and K. Dilcher, Pairs of reciprocal quadratic congruences involving primes, Fib. Quart. 51 (2) (2013) 98, after Theorem 3.
• Eric Weisstein's World of Mathematics, Pell Number
• Eric Weisstein's World of Mathematics, Integer Sequence Primes

Crossrefs

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

Cf. A005850.

Programs

Mathematica

lst={}; a=1; b=1; Do[c=a+2b; a=b; b=c; If[PrimeQ[c], AppendTo[lst, n]], {n, 2, 10000}]; lst
(* Second program: *)
Do[If[PrimeQ[Expand[((1 + Sqrt[2])^n + (1 - Sqrt[2])^n)/2]], Print[n]], {n, 0, 1000}] (* Artur Jasinski, Dec 10 2006 *)

PARI

isok(n) = isprime(polchebyshev(n, 1, I)/I^n); \\ Michel Marcus, Dec 07 2018

Extensions

a(24) from Eric W. Weisstein, May 22 2006

a(25) from Eric W. Weisstein, Aug 29 2006

a(26) from Eric W. Weisstein, Nov 11 2006

a(27) from Eric W. Weisstein, Nov 26 2006

a(28) from Eric W. Weisstein, Dec 10 2006

a(29) from Eric W. Weisstein, Jan 25 2007

a(30) from Robert Price, Dec 07 2018

a(31) from Robert Price, Dec 05 2023