A319040 Numbers k > 1 such that Pell(k) == 1 (mod k).

7, 17, 23, 31, 35, 41, 47, 71, 73, 79, 89, 97, 103, 113, 127, 137, 151, 167, 169, 191, 193, 199, 223, 233, 239, 241, 257, 263, 271, 281, 311, 313, 337, 353, 359, 367, 383, 385, 401, 409, 431, 433, 439, 449, 457, 463, 479, 487, 503, 521, 569, 577, 593, 599

Offset

1,1

Comments

It appears that most of the terms of this sequence are primes. The composite terms are 35, 169, 385, 899, 961, 1121, ... (A319042).

The primes in the sequence give A001132 (primes == +-1 (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

Links

Table of n, a(n) for n=1..54.

Example

k = 7 is in the sequence since Pell(7) = 169 = 7 * 24 + 1 == 1 (mod 7).
k = 11 is not in the sequence: Pell(11) = 5741 = 11 * 522 - 1 !== 1 (mod 11).
k = 35 is in the sequence: Pell(35) = 8822750406821 = 35 * 252078583052 + 1 == 1 (mod 35).

Maple

isA319040 := k -> simplify(2^(k-1)*hypergeom([1-k/2, (1-k)/2], [1-k], -1)) mod k = 1: A319040List := b -> select(isA319040, [$1..b]):
A319040List(600); # Peter Luschny, Sep 09 2018

Mathematica

Select[Range[500], Mod[Fibonacci[#, 2], #] == 1 &] (* Alonso del Arte, Sep 08 2018 *)

Crossrefs

Cf. A000129 (Pell numbers), A001132, A023173, A319041, A319042, A319043.

Sequence in context: A032454 A107643 A289363 * A216838 A198441 A058529

Adjacent sequences: A319037 A319038 A319039 * A319041 A319042 A319043

Keyword

nonn

Author

Jon E. Schoenfield, Sep 08 2018

Status

approved