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

Author

Jon E. Schoenfield, Sep 08 2018

Keywords

• nonn

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

Examples

			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).
		

Links

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

Crossrefs

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

Programs

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 *)