A319041 - OEIS

%I #43 Feb 03 2019 09:43:52

%S 3,5,11,13,19,29,37,43,53,59,61,67,83,101,107,109,131,139,149,157,163,

%T 173,179,181,197,211,227,229,251,269,277,283,293,307,317,331,347,349,

%U 373,379,389,397,419,421,443,461,467,491,499,509,523,541,547,557

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

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

%C 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

%C It appears that this sequence is (A042999 \ {2}) UNION A319043. - _Georg Fischer_, Oct 17 2018

%H Seiichi Manyama, <a href="/A319041/b319041.txt">Table of n, a(n) for n = 1..9999</a> (offset adapted by _Georg Fischer_, Jan 31 2019)

%e k = 3 is in the sequence since Pell(3) = 5 = 3*2 - 1 == -1 (mod 3).

%e k = 7 is not in the sequence: Pell(7) = 169 = 7*24 + 1 !== -1 (mod 7).

%t Select[Range[800], Mod[Fibonacci[-#, 2], -#]== -1 &] (* _Vincenzo Librandi_, Sep 09 2018 after _Alonso del Arte_; {1} removed by _Georg Fischer_, Jan 31 2019 *)

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

%K nonn

%O 1,1

%A _Jon E. Schoenfield_, Sep 08 2018