A099011 Pell pseudoprimes: odd composite numbers n such that P(n)-Kronecker(2,n) is divisible by n.

169, 385, 741, 961, 1121, 2001, 3827, 4879, 5719, 6215, 6265, 6441, 6479, 6601, 7055, 7801, 8119, 9799, 10945, 11395, 13067, 13079, 13601, 15841, 18241, 19097, 20833, 20951, 24727, 27839, 27971, 29183, 29953, 31417, 31535, 34561, 35459, 37345

Offset

1,1

Comments

Here P(n) are the Pell numbers (A000129), defined by P(0)=0, P(1)=1, P(x) = 2*P(x-1) + P(x-2) and Kronecker(2,n) is equal to 1 if n is congruent to +-1 mod 8 and equal to -1 if n is congruent to +-3 mod 8.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000 (from Dana Jacobsen's site, terms 1..200 from Ralf Stephan)

Antonio J. Di Scala, Nadir Murru, and Carlo Sanna, Lucas pseudoprimes and the Pell conic, arXiv:2001.00353 [math.NT], 2020.

Dana Jacobsen, Pseudoprime Statistics, Tables, and Data (includes terms to 5e12)

Example

169 is a Pell pseudoprime because P(169)-Kronecker(2,169) is divisible by 169.

Mathematica

pell[0] = 0; pell[1] = 1; pell[n_] := pell[n] = 2*pell[n - 1] + pell[n - 2]; pellpspQ[n_] := OddQ[n] && CompositeQ[n] && Divisible[pell[n] - JacobiSymbol[2, n], n]; Select[Range[40000], pellpspQ] (* Amiram Eldar, Nov 22 2019 *)

Prog

(Perl)
use Math::Prime::Util qw/:all/;
my($U, $V);
foroddcomposites {
  ($U, $V) = lucas_sequence($_, 2, -1, $_);
  $U = ($U - kronecker(2, $_)) % $_;
  print "$_\n" if $U == 0;
} 1e11; # Dana Jacobsen, Sep 13 2014
(PARI) genit(nterms=100)={my(arr=List(), q, z); forcomposite(n=9, +oo, if(n%2==0, next); q=([2, 1; 1, 0]^n)[2, 1]; z=(q-kronecker(2, n))%n; if(z==0, listput(arr, n)); if(#arr>=nterms, break)); arr} \\ Bill McEachen, Jun 24 2023 (incorporates A000129 code of Greathouse)

Crossrefs

Cf. A000129.

Sequence in context: A325421 A296304 A156159 * A330276 A351337 A327652

Adjacent sequences: A099008 A099009 A099010 * A099012 A099013 A099014

Keyword

nonn

Author

Jack Brennen, Nov 13 2004

Status

approved