A096650 Indices of prime Pell numbers.

2, 3, 5, 11, 13, 29, 41, 53, 59, 89, 97, 101, 167, 181, 191, 523, 929, 1217, 1301, 1361, 2087, 2273, 2393, 8093, 13339, 14033, 23747, 28183, 34429, 36749, 90197

Offset: 1

Author

Julien Peter Benney (jpbenney(AT)ftml.net), Aug 15 2004

Keywords

• nonn
• hard
• more

Comments

For a Pell number to be prime, the index must be prime. The indices greater than 523 yield probable primes. No others less than 100000. - T. D. Noe, Sep 13 2004

n divides m if and only if A000129(n) divides A000129(m). This is the reason of the fact that this sequence is a subsequence of A000040. For complement of this sequence see A270387. - Altug Alkan, Apr 29 2016

Examples

			P(11)=5741, which is prime.
		

Links

• Steve Fan and Paul Pollack, Extremal elasticity of quadratic orders, arXiv:2503.07801 [math.NT], 2025. See p. 17.
• J. L. Schiffman, Exploring the Fibonacci sequence of order two with CAS technology, Paper C027, Electronic Proceedings of the Twenty-fourth Annual International Conference on Technology in Collegiate Mathematics, Orlando, Florida, March 22-25, 2012. See p. 262. - _N. J. A. Sloane_, Mar 27 2014
• Eric Weisstein's World of Mathematics, Pell Number.
• Eric Weisstein's World of Mathematics, Integer Sequence Primes .

Crossrefs

Cf. A000129 (Pell numbers), A086383 (prime Pell numbers), A270387.

Programs

Maple

Pell:= gfun:-rectoproc( {a(0) = 0, a(1) = 1, a(n) = 2*a(n-1) + a(n-2)},a(n), remember):
select(t -> isprime(t) and isprime(Pell(t)), [2, seq(2*i+1, i=1..2000)]); # Robert Israel, Aug 28 2015

Mathematica

lst={}; a=0; b=1; Do[c=a+2b; a=b; b=c; If[PrimeQ[c], AppendTo[lst, n]], {n, 2, 10000}]; lst (* T. D. Noe, Aug 17 2004 *)
Flatten@ Position[#, p_ /; PrimeQ@ p] - 1 &@ CoefficientList[Series[x/(1 - 2 x - x^2), {x, 0, 5000}], x] (* Michael De Vlieger, Apr 29 2016, after Stefan Steinerberger at A000129 *)

Extensions

More terms from T. D. Noe, Aug 17 2004

Further terms from T. D. Noe, Sep 13 2004