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

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Comments

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

References

• 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.

Links

• 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

Crossrefs

Cf. A000032, A002808.

Cf. A094394, A094395 (analogous numbers with the Fibonacci sequence). - Robert FERREOL, Jul 14 2015

Cf. A213060 (L(n) mod n).

Programs

Haskell

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

Maple

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

Mathematica

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

PARI

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

Python

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

Extensions

More terms from David Broadhurst