

A001606


Indices of prime Lucas numbers.
(Formerly M0961 N0358)


40



0, 2, 4, 5, 7, 8, 11, 13, 16, 17, 19, 31, 37, 41, 47, 53, 61, 71, 79, 113, 313, 353, 503, 613, 617, 863, 1097, 1361, 4787, 4793, 5851, 7741, 8467, 10691, 12251, 13963, 14449, 19469, 35449, 36779, 44507, 51169, 56003, 81671, 89849, 94823, 140057, 148091, 159521, 183089, 193201, 202667, 344293, 387433, 443609, 532277, 574219, 616787, 631181, 637751, 651821, 692147, 901657, 1051849
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OFFSET

1,2


COMMENTS

Some of the larger entries may only correspond to probable primes.
Since (as noted under A000032) L(n) divides L(mn) whenever m is odd, L(n) cannot be prime unless n is itself prime, or else n contains no odd divisor, i.e., is a power of 2. Potential divisors of L(n) must satisfy certain linear forms dependent upon the parity of n, as shown in Vajda (1989), p. 82 (with a slight typographical error in the proof).  John Blythe Dobson, Oct 22 2007
Powers of 2 in this sequence are 2, 4, 8, 16; for 5 <= m <= 24, L(2^m) is composite; no factors of L(2^m) are known for m = 25, 26, 27, 29, 32, 33... (See Link section).  Serge Batalov, May 30 2017
2316773 is in the sequence, but its position is not yet defined. L(2316773) is a 484177digit PRP.  Serge Batalov, Jun 11 2017


REFERENCES

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
S. Vajda, Fibonacci and Lucas numbers and the Golden Section: Theory and Applications. Chichester: Ellis Horwood Ltd., 1989.


LINKS

Table of n, a(n) for n=1..64.
J. Brillhart, P. L. Montgomery and R. D. Silverman, Tables of Fibonacci and Lucas factorizations, Math. Comp. 50 (1988), 251260.
D. Broadhurst, Lucas record follows Fibonacci, Yahoo! group "primenumbers", Apr 26 2001
D. Broadhurst, Lucas record follows Fibonacci [Cached copy]
C. K. Caldwell, The Prime Glossary, Lucas prime
H. Dubner and W. Keller, New Fibonacci and Lucas Primes, Math. Comp. 68 (1999) 417427.
Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 36.
Blair Kelly, Factorizations of Lucas numbers
Alex Kontorovich and Jeff Lagarias, On Toric Orbits in the Affine Sieve, arXiv:1808.03235 [math.NT], 2018.
H. Lifchitz and R. Lifchitz, PRP Top Records, L(n)
Mersenneforum, A collection of factors of L(2^m).
Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci nstep and Lucas nstep Sequences, J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4.
The Prime Database, V(81671)
Lawrence Somer and Michal Křížek, On Primes in Lucas Sequences, Fibonacci Quart. 53 (2015), no. 1, 223.
Eric Weisstein's World of Mathematics, Lucas Number.
Eric Weisstein's World of Mathematics, Integer Sequence Primes.


MATHEMATICA

Reap[For[k = 0, k < 20000, k++, If[PrimeQ[LucasL[k]], Print[k]; Sow[k]]] ][[2, 1]] (* JeanFrançois Alcover, Feb 27 2016 *)


PROG

(PARI) is(n)=ispseudoprime(fibonacci(n1)+fibonacci(n+1)) \\ Charles R Greathouse IV, Apr 24 2015


CROSSREFS

Cf. A000032, A000204, A001605, A005479.
Cf. A080327 (n for which Lucas(n) and Fibonacci(n) are both prime).
Sequence in context: A022559 A049781 A076697 * A014554 A114147 A281276
Adjacent sequences: A001603 A001604 A001605 * A001607 A001608 A001609


KEYWORD

nonn,hard,nice


AUTHOR

N. J. A. Sloane


EXTENSIONS

4 more terms from David Broadhurst, Jun 08 2001
More terms from T. D. Noe, Feb 15 2003 and Mar 04 2003; see link to The Prime Glossary.
387433, 443609, 532277 and 574219 found by Renaud Lifchitz, contributed by Eric W. Weisstein, Nov 29 2005
616787, 631181, 637751, 651821, 692147 found by Henri Lifchitz, circa Oct 01 2008, contributed by Alexander Adamchuk, Nov 28 2008
901657 and 1051849 found by Renaud Lifchitz, circa Nov 2008 and Mar 2009, contributed by Alexander Adamchuk, May 15 2010
1 more term from Serge Batalov, Jun 11 2017


STATUS

approved



