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

%I A005845 M5469 #100 Apr 06 2025 11:30:27
%S A005845 705,2465,2737,3745,4181,5777,6721,10877,13201,15251,24465,29281,
%T A005845 34561,35785,51841,54705,64079,64681,67861,68251,75077,80189,90061,
%U A005845 96049,97921,100065,100127,105281,113573,118441,146611,161027
%N A005845 Bruckman-Lucas pseudoprimes: k | (L_k - 1), where k is composite and L_k = Lucas numbers A000032.
%C A005845 This uses the definition of "Lucas pseudoprime" by Bruckman, not the one by Baillie and Wagstaff. - _R. J. Mathar_, Jul 15 2012
%C A005845 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
%C A005845 k in A002808 such that A213060(k) = 1. - _Robert Israel_, Jul 14 2015
%D A005845 Paulo Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 104.
%D A005845 Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 105.
%D A005845 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A005845 Somer, Lawrence. "Generalization of a Theorem of Bruckman on Dickson Pseudoprimes." Fibonacci Quarterly 60:4 (2022), 357-361.
%H A005845 Amiram Eldar, <a href="/A005845/b005845.txt">Table of n, a(n) for n = 1..10000</a> (from Dana Jacobsen's site, terms 1..1000 from T. D. Noe)
%H A005845 Dorin Andrica and Ovidiu Bagdasar, <a href="https://doi.org/10.1007/978-3-030-51502-7">Recurrent Sequences: Key Results, Applications, and Problems</a>, Springer (2020), p. 88.
%H A005845 Dorin Andrica and Ovidiu Bagdasar, <a href="https://doi.org/10.3390/math9080838">On Generalized Lucas Pseudoprimality of Level k</a>, Mathematics (2021) Vol. 9, 838.
%H A005845 R. Baillie and S. S. Wagstaff, <a href="http://dx.doi.org/10.1090/S0025-5718-1980-0583518-6">Lucas pseudoprimes</a>, Math. Comp 35 (1980) 1391-1417.
%H A005845 P. S. Bruckman, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/32-2/bruckman2.pdf">Lucas Pseudoprimes are odd</a>, Fib. Quart. 32 (1994), 155-157.
%H A005845 Dana Jacobsen, <a href="http://ntheory.org/pseudoprimes.html">Pseudoprime Statistics, Tables, and Data</a>.
%H A005845 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LucasPseudoprime.html">Lucas Pseudoprime</a>.
%H A005845 <a href="/index/Ps#pseudoprimes">Index entries for sequences related to pseudoprimes</a>
%p A005845 with(combinat):lucas:=n->fibonacci(n-1)+fibonacci(n+1):
%p A005845 test:=n->lucas(n) mod n=1:select(test and not isprime,[seq(n,n=1..10000)]); # _Robert FERREOL_, Jul 14 2015
%t A005845 Select[Range[2,170000],!PrimeQ[#]&&Divisible[LucasL[#]-1,#]&] (* _Harvey P. Dale_, Mar 08 2014 *)
%o A005845 (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
%o A005845 (Haskell)
%o A005845 a005845 n = a005845_list !! (n-1)
%o A005845 a005845_list = filter (\x -> (a000032 x - 1) `mod` x == 0) a002808_list
%o A005845 -- _Reinhard Zumkeller_, Nov 13 2014
%o A005845 (Python)
%o A005845 from sympy import isprime
%o A005845 from itertools import count, islice
%o A005845 def agen(): # generator of terms
%o A005845     L0, L1 = 2, 1
%o A005845     for k in count(1):
%o A005845         L0, L1 = L1, L0+L1
%o A005845         if k > 1 and not isprime(k) and (L0-1)%k == 0:
%o A005845             yield k
%o A005845 print(list(islice(agen(), 32))) # _Michael S. Branicky_, Apr 07 2024
%Y A005845 Cf. A000032, A002808.
%Y A005845 Cf. A094394, A094395 (analogous numbers with the Fibonacci sequence). - _Robert FERREOL_, Jul 14 2015
%Y A005845 Cf. A213060 (L(n) mod n).
%K A005845 nonn,nice
%O A005845 1,1
%A A005845 _N. J. A. Sloane_
%E A005845 More terms from _David Broadhurst_