login
A319040
Numbers k > 1 such that Pell(k) == 1 (mod k).
3
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
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
KEYWORD
nonn
AUTHOR
Jon E. Schoenfield, Sep 08 2018
STATUS
approved