A319041 Numbers k > 1 such that Pell(k) == -1 (mod k).

3, 5, 11, 13, 19, 29, 37, 43, 53, 59, 61, 67, 83, 101, 107, 109, 131, 139, 149, 157, 163, 173, 179, 181, 197, 211, 227, 229, 251, 269, 277, 283, 293, 307, 317, 331, 347, 349, 373, 379, 389, 397, 419, 421, 443, 461, 467, 491, 499, 509, 523, 541, 547, 557

Offset

1,1

Comments

It appears that most of the terms of this sequence are primes. The composite terms are 741, 3827, 11395, 13067, 27971, ... (A319043).

The primes in the sequence give A003629 (primes == +-3 (mod 8)), since for primes p we have Pell(p) == (2/p) (mod p) where (2/p) is the Legendre symbol. - Jianing Song, Sep 10 2018

It appears that this sequence is (A042999 \ {2}) UNION A319043. - Georg Fischer, Oct 17 2018

Links

Seiichi Manyama, Table of n, a(n) for n = 1..9999 (offset adapted by Georg Fischer, Jan 31 2019)

Example

k = 3 is in the sequence since Pell(3) = 5 = 3*2 - 1 == -1 (mod 3).
k = 7 is not in the sequence: Pell(7) = 169 = 7*24 + 1 !== -1 (mod 7).

Mathematica

Select[Range[800], Mod[Fibonacci[-#, 2], -#]== -1 &] (* Vincenzo Librandi, Sep 09 2018 after Alonso del Arte; {1} removed by Georg Fischer, Jan 31 2019 *)

Crossrefs

Cf. A000129 (Pell numbers), A003629, A042999, A319040, A319042, A319043.

Sequence in context: A059350 A059644 A059646 * A003629 A175865 A001122

Adjacent sequences: A319038 A319039 A319040 * A319042 A319043 A319044

Keyword

nonn

Author

Jon E. Schoenfield, Sep 08 2018

Status

approved