CINXE.COM
A001606 - OEIS
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 3.2 Final//EN"> <html> <head> <link rel="stylesheet" href="/styles.css"> <meta name="format-detection" content="telephone=no"> <meta http-equiv="content-type" content="text/html; charset=utf-8"> <meta name=viewport content="width=device-width, initial-scale=1"> <meta name="keywords" content="OEIS,integer sequences,Sloane" /> <title>A001606 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A001606" function redir() { var host = document.location.hostname; if(host != "oeis.org" && host != "127.0.0.1" && !/^([0-9.]+)$/.test(host) && host != "localhost" && host != "localhost.localdomain") { document.location = "https"+":"+"//"+"oeis"+".org/" + myURL; } } function sf() { if(document.location.pathname == "/" && document.f) document.f.q.focus(); } </script> </head> <body bgcolor=#ffffff onload="redir();sf()"> <div class=loginbar> <div class=login> <a href="/login?redirect=%2fA001606">login</a> </div> </div> <div class=center><div class=top> <center> <div class=donors> The OEIS is supported by <a href="http://oeisf.org/#DONATE">the many generous donors to the OEIS Foundation</a>. </div> <div class=banner> <a href="/"><img class=banner border="0" width="600" src="/banner2021.jpg" alt="A001606 - OEIS"></a> </div> </center> </div></div> <div class=center><div class=pagebody> <div class=searchbarcenter> <form name=f action="/search" method="GET"> <div class=searchbargreet> <div class=searchbar> <div class=searchq> <input class=searchbox maxLength=1024 name=q value="" title="Search Query"> </div> <div class=searchsubmit> <input type=submit value="Search" name=go> </div> <div class=hints> <span class=hints><a href="/hints.html">Hints</a></span> </div> </div> <div class=searchgreet> (Greetings from <a href="/welcome">The On-Line Encyclopedia of Integer Sequences</a>!) </div> </div> </form> </div> <div class=sequence> <div class=space1></div> <div class=line></div> <div class=seqhead> <div class=seqnumname> <div class=seqnum> A001606 </div> <div class=seqname> Indices of prime Lucas numbers. <br><font size=-1>(Formerly M0961 N0358)</font> </div> </div> <div class=scorerefs> 44 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>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</div> <div class=seqdatalinks> (<a href="/A001606/list">list</a>; <a href="/A001606/graph">graph</a>; <a href="/search?q=A001606+-id:A001606">refs</a>; <a href="/A001606/listen">listen</a>; <a href="/history?seq=A001606">history</a>; <a href="/search?q=id:A001606&fmt=text">text</a>; <a href="/A001606/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>1,2</div> </div> </div> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <div class=sectline>Some of the larger entries may only correspond to probable primes.</div> <div class=sectline>Since (as noted under <a href="/A000032" title="Lucas numbers beginning at 2: L(n) = L(n-1) + L(n-2), L(0) = 2, L(1) = 1.">A000032</a>) 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). - <a href="/wiki/User:John_Blythe_Dobson">John Blythe Dobson</a>, Oct 22 2007</div> <div class=sectline>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). - <a href="/wiki/User:Serge_Batalov">Serge Batalov</a>, May 30 2017</div> <div class=sectline>2316773 is in the sequence, but its position is not yet defined. L(2316773) is a 484177-digit PRP. - <a href="/wiki/User:Serge_Batalov">Serge Batalov</a>, Jun 11 2017</div> </div> </div> <div class=section> <div class=sectname>REFERENCES</div> <div class=sectbody> <div class=sectline>N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).</div> <div class=sectline>N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).</div> <div class=sectline>S. Vajda, Fibonacci and Lucas numbers and the Golden Section: Theory and Applications. Chichester: Ellis Horwood Ltd., 1989.</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline><a href="/A001606/b001606.txt">Table of n, a(n) for n=1..64.</a></div> <div class=sectline>J. Brillhart, P. L. Montgomery and R. D. Silverman, <a href="http://dx.doi.org/10.1090/S0025-5718-1988-0917832-6">Tables of Fibonacci and Lucas factorizations</a>, Math. Comp. 50 (1988), 251-260.</div> <div class=sectline>D. Broadhurst, <a href="http://groups.yahoo.com/group/primenumbers/message/1003">Lucas record follows Fibonacci</a>, Yahoo! group "primenumbers", Apr 26 2001</div> <div class=sectline>D. Broadhurst, <a href="/A001606/a001606.png">Lucas record follows Fibonacci</a> [Cached copy]</div> <div class=sectline>C. K. Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/page.php?sort=LucasPrime">Lucas prime</a></div> <div class=sectline>H. Dubner and W. Keller, <a href="http://dx.doi.org/10.1090/S0025-5718-99-00981-3">New Fibonacci and Lucas Primes</a>, Math. Comp. 68 (1999) 417-427.</div> <div class=sectline>Dov Jarden, <a href="/A001602/a001602.pdf">Recurring Sequences</a>, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 36.</div> <div class=sectline>Blair Kelly, <a href="http://mersennus.net/fibonacci//lucas.txt">Factorizations of Lucas numbers</a></div> <div class=sectline>Alex Kontorovich and Jeff Lagarias, <a href="https://arxiv.org/abs/1808.03235">On Toric Orbits in the Affine Sieve</a>, arXiv:1808.03235 [math.NT], 2018.</div> <div class=sectline>H. Lifchitz and R. Lifchitz, <a href="http://www.primenumbers.net/prptop/searchform.php?form=L(n)">PRP Top Records, L(n)</a></div> <div class=sectline>Mersenneforum, <a href="http://www.mersenneforum.org/showthread.php?p=459964#post459964">A collection of factors of L(2^m)</a>.</div> <div class=sectline>Tony D. Noe and Jonathan Vos Post, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Noe/noe5.html">Primes in Fibonacci n-step and Lucas n-step Sequences,</a> J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4.</div> <div class=sectline>The Prime Database, <a href="https://t5k.org/primes/page.php?id=115420"> V(81671)</a></div> <div class=sectline>Lawrence Somer and Michal K艡铆啪ek, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/53-1/SomerKrizek5222014.pdf">On Primes in Lucas Sequences</a>, Fibonacci Quart. 53 (2015), no. 1, 2-23.</div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LucasNumber.html">Lucas Number</a>.</div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IntegerSequencePrimes.html">Integer Sequence Primes</a>.</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>Reap[For[k = 0, k < 20000, k++, If[PrimeQ[LucasL[k]], Print[k]; Sow[k]]] ][[2, 1]] (* <a href="/wiki/User:Jean-Fran莽ois_Alcover">Jean-Fran莽ois Alcover</a>, Feb 27 2016 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(PARI) is(n)=ispseudoprime(fibonacci(n-1)+fibonacci(n+1)) \\ <a href="/wiki/User:Charles_R_Greathouse_IV">Charles R Greathouse IV</a>, Apr 24 2015</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A000032" title="Lucas numbers beginning at 2: L(n) = L(n-1) + L(n-2), L(0) = 2, L(1) = 1.">A000032</a>, <a href="/A000204" title="Lucas numbers (beginning with 1): L(n) = L(n-1) + L(n-2) with L(1) = 1, L(2) = 3.">A000204</a>, <a href="/A001605" title="Indices of prime Fibonacci numbers.">A001605</a>, <a href="/A005479" title="Prime Lucas numbers (cf. A000032).">A005479</a>.</div> <div class=sectline>Cf. <a href="/A080327" title="Numbers k for which Lucas(k) and Fibonacci(k) are both prime.">A080327</a> (n for which Lucas(n) and Fibonacci(n) are both prime).</div> <div class=sectline>Sequence in context: <a href="/A022559" title="Sum of exponents in prime-power factorization of n!.">A022559</a> <a href="/A049781" title="a(n) = (b(n) + 1)/2, where b = A000959 (the lucky numbers).">A049781</a> <a href="/A076697" title="Next-to-largest factor of Lucas(n).">A076697</a> * <a href="/A014554" title="Erroneous version of A001606.">A014554</a> <a href="/A114147" title="Self-describing sequence : 1 prime between two nonprimes, then 2 primes between two nonprimes, then 4 primes, then 5, then 7...">A114147</a> <a href="/A281276" title="Numbers k such that (2*10^k + 457)/9 is prime.">A281276</a></div> <div class=sectline>Adjacent sequences: <a href="/A001603" title="Odd-indexed terms of A124296.">A001603</a> <a href="/A001604" title="Odd-indexed terms of A124297.">A001604</a> <a href="/A001605" title="Indices of prime Fibonacci numbers.">A001605</a> * <a href="/A001607" title="a(n) = -a(n-1) - 2*a(n-2).">A001607</a> <a href="/A001608" title="Perrin sequence (or Ondrej Such sequence): a(n) = a(n-2) + a(n-3) with a(0) = 3, a(1) = 0, a(2) = 2.">A001608</a> <a href="/A001609" title="a(1) = a(2) = 1, a(3) = 4; thereafter a(n) = a(n-1) + a(n-3).">A001609</a></div> </div> </div> <div class=section> <div class=sectname>KEYWORD</div> <div class=sectbody> <div class=sectline><span title="a sequence of nonnegative numbers">nonn</span>,<span title="next term not known, may be hard to find. please extend this sequence">hard</span>,<span title="an exceptionally nice sequence">nice</span>,<span title="edited within the last two weeks">changed</span></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:N._J._A._Sloane">N. J. A. Sloane</a></div> </div> </div> <div class=section> <div class=sectname>EXTENSIONS</div> <div class=sectbody> <div class=sectline>4 more terms from <a href="/wiki/User:David_Broadhurst">David Broadhurst</a>, Jun 08 2001</div> <div class=sectline>More terms from <a href="/wiki/User:T._D._Noe">T. D. Noe</a>, Feb 15 2003 and Mar 04 2003; see link to The Prime Glossary.</div> <div class=sectline>387433, 443609, 532277 and 574219 found by Renaud Lifchitz, contributed by <a href="/wiki/User:Eric_W._Weisstein">Eric W. Weisstein</a>, Nov 29 2005</div> <div class=sectline>616787, 631181, 637751, 651821, 692147 found by <a href="/wiki/User:Henri_Lifchitz">Henri Lifchitz</a>, circa Oct 01 2008, contributed by <a href="/wiki/User:Alexander_Adamchuk">Alexander Adamchuk</a>, Nov 28 2008</div> <div class=sectline>901657 and 1051849 found by Renaud Lifchitz, circa Nov 2008 and Mar 2009, contributed by <a href="/wiki/User:Alexander_Adamchuk">Alexander Adamchuk</a>, May 15 2010</div> <div class=sectline>1 more term from <a href="/wiki/User:Serge_Batalov">Serge Batalov</a>, Jun 11 2017</div> </div> </div> <div class=section> <div class=sectname>STATUS</div> <div class=sectbody> <div class=sectline>approved</div> </div> </div> </div> <div class=space10></div> </div> </div></div> <p> <div class=footerpad></div> <div class=footer> <center> <div class=bottom> <div class=linksbar> <a href="/">Lookup</a> <a href="/wiki/Welcome"><font color="red">Welcome</font></a> <a href="/wiki/Main_Page"><font color="red">Wiki</font></a> <a href="/wiki/Special:RequestAccount">Register</a> <a href="/play.html">Music</a> <a href="/plot2.html">Plot 2</a> <a href="/demo1.html">Demos</a> <a href="/wiki/Index_to_OEIS">Index</a> <a href="/webcam">WebCam</a> <a href="/Submit.html">Contribute</a> <a href="/eishelp2.html">Format</a> <a href="/wiki/Style_Sheet">Style Sheet</a> <a href="/transforms.html">Transforms</a> <a href="/ol.html">Superseeker</a> <a href="/recent">Recents</a> </div> <div class=linksbar> <a href="/community.html">The OEIS Community</a> </div> <div class=linksbar> Maintained by <a href="http://oeisf.org">The OEIS Foundation Inc.</a> </div> <div class=dbinfo>Last modified February 17 17:38 EST 2025. Contains 380975 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>