A217120 -id:A217120 - cp's OEIS frontend

A005845 Bruckman-Lucas pseudoprimes: k | (L_k - 1), where k is composite and L_k = Lucas numbers A000032.

705, 2465, 2737, 3745, 4181, 5777, 6721, 10877, 13201, 15251, 24465, 29281, 34561, 35785, 51841, 54705, 64079, 64681, 67861, 68251, 75077, 80189, 90061, 96049, 97921, 100065, 100127, 105281, 113573, 118441, 146611, 161027

Offset: 1

N. J. A. Sloane

This uses the definition of "Lucas pseudoprime" by Bruckman, not the one by Baillie and Wagstaff. - R. J. Mathar, Jul 15 2012
Unlike the earlier Baillie-Wagstaff Lucas pseudoprimes A217120, these have significant overlap with the Fermat primality test. For example, the number 82380774001 is both an A005845 Lucas pseudoprime and a Fermat pseudoprime to the first 407 prime bases. - Dana Jacobsen, Jan 10 2015
k in A002808 such that A213060(k) = 1. - Robert Israel, Jul 14 2015

Paulo Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 104.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 105.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Somer, Lawrence. "Generalization of a Theorem of Bruckman on Dickson Pseudoprimes." Fibonacci Quarterly 60:4 (2022), 357-361.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (from Dana Jacobsen's site, terms 1..1000 from T. D. Noe)
Dorin Andrica and Ovidiu Bagdasar, Recurrent Sequences: Key Results, Applications, and Problems, Springer (2020), p. 88.
Dorin Andrica and Ovidiu Bagdasar, On Generalized Lucas Pseudoprimality of Level k, Mathematics (2021) Vol. 9, 838.
R. Baillie and S. S. Wagstaff, Lucas pseudoprimes, Math. Comp 35 (1980) 1391-1417.
P. S. Bruckman, Lucas Pseudoprimes are odd, Fib. Quart. 32 (1994), 155-157.
Dana Jacobsen, Pseudoprime Statistics, Tables, and Data.
Eric Weisstein's World of Mathematics, Lucas Pseudoprime.
Index entries for sequences related to pseudoprimes

Cf. A000032, A002808.
Cf. A094394, A094395 (analogous numbers with the Fibonacci sequence). - Robert FERREOL, Jul 14 2015
Cf. A213060 (L(n) mod n).

a005845 n = a005845_list !! (n-1)
a005845_list = filter (\x -> (a000032 x - 1) `mod` x == 0) a002808_list
-- Reinhard Zumkeller, Nov 13 2014

with(combinat):lucas:=n->fibonacci(n-1)+fibonacci(n+1):
test:=n->lucas(n) mod n=1:select(test and not isprime,[seq(n,n=1..10000)]); # Robert FERREOL, Jul 14 2015

Select[Range[2,170000],!PrimeQ[#]&&Divisible[LucasL[#]-1,#]&] (* Harvey P. Dale, Mar 08 2014 *)

is(n)=my(M=Mod([1,1;1,0],n)^n);M[1,1]+M[2,2]==1 && !isprime(n) && n>1 \\ Charles R Greathouse IV, Dec 27 2013

from sympy import isprime
from itertools import count, islice
def agen(): # generator of terms
    L0, L1 = 2, 1
    for k in count(1):
        L0, L1 = L1, L0+L1
        if k > 1 and not isprime(k) and (L0-1)%k == 0:
            yield k
print(list(islice(agen(), 32))) # Michael S. Branicky, Apr 07 2024

More terms from David Broadhurst

A217255 Strong Lucas pseudoprimes.

Original entry on oeis.org

5459, 5777, 10877, 16109, 18971, 22499, 24569, 25199, 40309, 58519, 75077, 97439, 100127, 113573, 115639, 130139, 155819, 158399, 161027, 162133, 176399, 176471, 189419, 192509, 197801, 224369, 230691, 231703, 243629, 253259, 268349, 288919, 313499, 324899

Offset: 1

Robert Baillie, Mar 16 2013

nonn

Strong Lucas pseudoprimes with parameters (P, Q) defined by Selfridge's Method A.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (from Dana Jacobsen's site, terms 1..2000 from R. J. Mathar)
Martin R. Albrecht, Jake Massimo, Kenneth G. Paterson, Juraj Somorovsky, Prime and Prejudice: Primality Testing Under Adversarial Conditions, Proceedings of the 2018 ACM SIGSAC Conference on Computer and Communications Security, 281-298.
Robert Baillie and Samuel S. Wagstaff, Jr., Lucas Pseudoprimes, Mathematics of Computation, 35 (1980), 1391-1417.
Robert Baillie, Mathematica program for A217255
Dana Jacobsen, Pseudoprime Statistics, Tables, and Data (includes terms through 10^14)

Cf. A217120 (Lucas pseudoprimes as defined by Baillie and Wagstaff).
Cf. A005845 (Lucas pseudoprimes as defined by Bruckman).
Cf. A217719 (extra strong Lucas pseudoprimes as defined by Baillie).

Mathematica
```
(* see link *)
```

A217719 Extra strong Lucas pseudoprimes.

Original entry on oeis.org

989, 3239, 5777, 10877, 27971, 29681, 30739, 31631, 39059, 72389, 73919, 75077, 100127, 113573, 125249, 137549, 137801, 153931, 155819, 161027, 162133, 189419, 218321, 231703, 249331, 370229, 429479, 430127, 459191, 473891, 480689, 600059, 621781, 632249, 635627

Offset: 1

Robert Baillie, Mar 21 2013

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (from Dana Jacobsen's site)
Robert Baillie, Mathematica code for extra strong Lucas pseudoprimes
Jon Grantham, Frobenius Pseudoprimes, Mathematics of Computation, 7 (2000), 873-891.
Dana Jacobsen, Pseudoprime Statistics, Tables, and Data

Cf. A217120 (Lucas pseudoprimes as defined by Baillie and Wagstaff).

Cf. A217255 (strong Lucas pseudoprimes as defined by Baillie and Wagstaff).

use ntheory ":all"; foroddcomposites { say if is_extra_strong_lucas_pseudoprime($) } 1e10;  # _Dana Jacobsen, May 10 2015

A290560 Generalized Lucas-Carmichael numbers for D=9697.

Original entry on oeis.org

1, 35, 143, 323, 385, 455, 595, 665, 899, 935, 1045, 1295, 1547, 1729, 2639, 2737, 2821, 2915, 3289, 3689, 4355, 4465, 5005, 5183, 5291, 6479, 6721, 8855, 8911, 9215, 9361, 10153, 10439, 10465, 11305, 11663, 11951, 15841, 17119, 18095, 19981, 20909, 22607

Offset: 1

André Hernández-Espiet, Aug 06 2017

nonn

On the set Lc(Z/NZ,D) = {(x,y) in (Z/NZ)^2 : x^2 - Dy^2 = 1 (mod N)}, define an operation as follows: (x,y)x(z,w) = (xz+Dyw, xw+zy) (mod N). The set Lc(Z/NZ, D) endowed with this operation is a group. Moreover, the set of Lucas numbers endowed with this operation is a subgroup of Lc(Z/NZ, D).
The following results appear in Babinkostova, et al.: If q is a prime, then #Lc(Z/(q^e)Z, D) = (q-(D|q))q^(e-1).
The group Lc(Z/(q^e)Z, D) is cyclic for e > 0. This result was proven in Hinkel, 2007 for the case when e = 1. We showed that the statement is true for e > 1 (Babinkostova, et al.).
The following notions are introduced in Babinkostova, et al.: A composite integer N is a generalized Lucas pseudoprime (or Lucas pseudoprime in Babinkostova, et al.) to base P in Lc(Z/NZ, D) and integer D if (N-(D|N))P = O, where O is the identity of the group.
We define a composite integer N to be a generalized Lucas-Carmichael number if for all P in Lc(Z/NZ, D) it is true that (N-(D|N))P = O.
The following Korselt-like criterion holds for a generalized Lucas-Carmichael number: A composite number N is a generalized Lucas-Carmichael number if and only if N is squarefree and for every prime factor q of N, (q-(D|q)) divides (N-(D|N)).
This sequence is a list of generalized Lucas-Carmichael numbers for D=9697.
For prime values of D less than 10000 and odd nonprime values of N less than 1000000, this is the longest sequence of generalized Lucas-Carmichael numbers.
The resulting sequence of generalized Lucas-Carmichael numbers is based on work done by L. Babinkostova, B. Bentz, M. I. Hassan, and H. Kim.

			We will illustrate an example using the Korselt criterion for generalized Lucas pseudoprimes. Let us observe the second term, 35. Note that 35 = 5*7, so that it is squarefree. Now note that (5-(9697|5)) = 6 and (7-(9697|7)) = 6, both of which divide (35-(9697|35)) = 36. Therefore, by the Korselt criterion for generalized Lucas pseudoprimes, we have that 35 is a generalized Lucas Carmichael number for D = 9697.

L. Babinkostova, B. Bentz, M. Hassan, A. Hernández-Espiet, and H. J. Kim, Anomalous Primes and the Elliptic Korselt Criterion. (poster presentation)
R. Baillie and S. S. Wagstaff, Lucas Pseudoprimes, Mathematics of Computation, Vol. 35, (1980), 1391-1417.
D. M. Gordon and C. Pomerance, The distribution of Lucas and elliptic pseudoprimes, Mathematics of Computation, Vol. 57: 196, 825-838.
D. E. Hinkel, An investigation of Lucas sequences, Master's theses, Boise State University (2007).
J. Smith, Solvability characterizations of Pell like equations, Master's theses, Boise State University (2009).
Sage program that computes the terms of the sequence: Generalized Lucas Pseudoprime Program

Cf. A005845, A164824, A217120, A217255, A217719, A227905.

# A program in SageMath is given in the links section.

cp's OEIS Frontend

A005845 Bruckman-Lucas pseudoprimes: k | (L_k - 1), where k is composite and L_k = Lucas numbers A000032.

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A217255 Strong Lucas pseudoprimes.

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A217719 Extra strong Lucas pseudoprimes.

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A290560 Generalized Lucas-Carmichael numbers for D=9697.

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SageMath

A365514 Lucas-V pseudoprimes: composites c such that V_{c+1} == 2Q (mod c), where V_k is a Lucas sequence with parameters P and Q.

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