%I A086383 #64 Jun 05 2026 10:40:55
%S A086383 2,5,29,5741,33461,44560482149,1746860020068409,68480406462161287469,
%T A086383 13558774610046711780701,4125636888562548868221559797461449,
%U A086383 4760981394323203445293052612223893281,161733217200188571081311986634082331709,2964793555272799671946653940160950323792169332712780937764687561
%N A086383 Prime terms in the sequence of Pell numbers, A000129.
%C A086383 Previous Name: Primes found among the denominators of the continued fraction rational approximations to sqrt(2).
%C A086383 See A096650 for the indices. - _Jon E. Schoenfield_, Jan 25 2017
%C A086383 A056869 is essentially the same sequence. - _Jianing Song_, Jan 02 2019
%H A086383 Harvey P. Dale, <a href="/A086383/b086383.txt">Table of n, a(n) for n = 1..23</a>
%H A086383 J. L. Schiffman, <a href="https://web.archive.org/web/20150410161444/http://archives.math.utk.edu/ICTCM/i/24/C027.html">Exploring the Fibonacci sequence of order two with CAS technology</a>, 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
%H A086383 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IntegerSequencePrimes.html">Integer Sequence Primes</a>.
%H A086383 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PellPrime.html">Pell Prime</a>.
%F A086383 a(n) = A000129(A096650(n)). - _Jon E. Schoenfield_, Jan 25 2017
%F A086383 a(n) = A056869(n-1), n > 1. - _Jianing Song_, Jan 02 2019
%e A086383 a(1) = 2 = A000129(2), a(2) = 5 = A000129(3), a(3) = 29 = A000129(5), etc. - _Zak Seidov_, Oct 21 2013 [Corrected by _Jianing Song_, Jan 02 2019]
%t A086383 Join[{2},Select[Table[ChebyshevU[k, 3] - ChebyshevU[k - 1, 3], {k, 0, 50}], PrimeQ]] (* _Ed Pegg Jr_, May 10 2007 *)
%t A086383 (* Alternative: *)
%t A086383 Select[Denominator[Convergents[Sqrt[2], 150]],PrimeQ] (* _Harvey P. Dale_, Dec 19 2012 *)
%t A086383 (* Alternative: *)
%t A086383 Select[LinearRecurrence[{2, 1}, {0, 1}, 16], PrimeQ] (* _Zak Seidov_, Oct 21 2013 *)
%t A086383 (* Alternative: *)
%t A086383 Select[Fibonacci[Range[100], 2], PrimeQ] (* _Eric W. Weisstein_, Jun 05 2026 *)
%o A086383 (PARI) \\ Continued fraction rational approximation of numeric constants f. m=steps.
%o A086383 cfracdenomprime(m,f) = { default(realprecision,3000); cf = vector(m+10); x=f; for(n=0,m, i=floor(x); x=1/(x-i); cf[n+1] = i; ); for(m1=0,m, r=cf[m1+1]; forstep(n=m1,1,-1, r = 1/r; r+=cf[n]; ); numer=numerator(r); denom=denominator(r); if(ispseudoprime(denom),print1(denom,",")); ) }
%o A086383 (GAP) f:=[0,1];; for n in [3..100] do f[n]:=2*f[n-1]+f[n-2]; od; a:=Filtered(f,IsPrime);; Print(a); # _Muniru A Asiru_, Jan 03 2019
%Y A086383 Cf. A000129, A056869, A096650.
%K A086383 nonn
%O A086383 1,1
%A A086383 _Cino Hilliard_, Sep 06 2003; corrected Jul 30 2004
%E A086383 Name changed (to a Comments entry from _Zak Seidov_, Oct 21 2013) by _Jon E. Schoenfield_, Jan 26 2017