A319042 -id:A319042 - cp's OEIS frontend

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

Jon E. Schoenfield, Sep 08 2018

nonn

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

			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).

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

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

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

A319043 Composite numbers k such that Pell(k) == -1 (mod k).

Original entry on oeis.org

741, 3827, 11395, 13067, 27971, 35459, 39059, 84587, 92833, 117739, 134579, 134945, 155819, 177497, 189419, 332949, 382771, 437579, 469699, 473891, 548627, 600059, 632269, 643259, 656083, 677379, 724883, 783579, 828827, 895299, 966779, 1015429, 1021987

Offset: 1

Jon E. Schoenfield, Sep 08 2018

nonn

It appears that most of the terms of A319041 (Numbers k such that Pell(k) == -1 (mod k)) are primes; this sequence lists the composites.
For the composite numbers k such that Pell(k) == 1 (mod k), see A319042.
Numbers that are terms of this sequence seem to be considerably less common than those in A319042; e.g., the numbers of terms in that sequence up to 10^3, 10^4, 10^5, and 10^6 are 5, 21, 67, and 200, respectively, while the corresponding term counts here are only 1, 2, 9, and 31. Why is this?

			k=741 is in the sequence: Pell(741) = 741*M - 1 == -1 (mod 741) (where M is a large integer).
k=6 is not in the sequence: Pell(6) = 70 = 6*12 - 2 !== -1 (mod 6).

Seiichi Manyama, Table of n, a(n) for n = 1..50

Cf. A000129 (Pell numbers), A094395, A319040, A319041, A319042.

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

Original entry on oeis.org

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

Jon E. Schoenfield, Sep 08 2018

nonn

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

			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).

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

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

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

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

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A319043 Composite numbers k such that Pell(k) == -1 (mod k).

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

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