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A094133 - 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>A094133 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A094133" 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=%2fA094133">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="A094133 - 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> A094133 </div> <div class=seqname> Leyland primes: 3, together with primes of form x^y + y^x, for x > y > 1. </div> </div> <div class=scorerefs> 23 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>3, 17, 593, 32993, 2097593, 8589935681, 59604644783353249, 523347633027360537213687137, 43143988327398957279342419750374600193, 4318114567396436564035293097707729426477458833, 5052785737795758503064406447721934417290878968063369478337</div> <div class=seqdatalinks> (<a href="/A094133/list">list</a>; <a href="/A094133/graph">graph</a>; <a href="/search?q=A094133+-id:A094133">refs</a>; <a href="/A094133/listen">listen</a>; <a href="/history?seq=A094133">history</a>; <a href="/search?q=id:A094133&fmt=text">text</a>; <a href="/A094133/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>1,1</div> </div> </div> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <div class=sectline>Contains <a href="/A061119" title="Primes in the sequence n^2 + 2^n (A001580).">A061119</a> as a subsequence.</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>Charles R Greathouse IV and Hans Havermann (Charles R Greathouse IV to 49), <a href="/A094133/b094133.txt">Table of n, a(n) for n = 1..100</a></div> <div class=sectline>Ed Copeland and Brady Haran, <a href="https://www.youtube.com/watch?v=Lsu2dIr_c8k">Leyland Numbers</a>, Numberphile video (2014).</div> <div class=sectline>Hans Havermann, <a href="http://chesswanks.com/num/a094133.txt">Table of n (where known), Leyland index, number of digits in decimal representation, and (x,y) pair for all known solutions</a>.</div> <div class=sectline>Ernest G. Hibbs, <a href="https://www.proquest.com/openview/4012f0286b785cd732c78eb0fc6fce80">Component Interactions of the Prime Numbers</a>, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.</div> <div class=sectline>Paul Leyland, <a href="http://www.leyland.vispa.com/numth/primes/xyyx.htm">Primes and PRPs of the form x^y + y^x</a>.</div> <div class=sectline>Norman Luhn, <a href="https://pzktupel.de/Primetables/TableLeyland1.php">Leyland table, 1st kind</a>.</div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>2^1 + 1^2, 3^2 + 2^3, 9^2 + 2^9, 15^2 + 2^15, 21^2 + 2^21, 33^2 + 2^33, 24^5 + 5^24, 56^3 + 3^56, 32^15 + 15^32, 54^7 + 7^54, 38^33 + 33^38.</div> </div> </div> <div class=section> <div class=sectname>MAPLE</div> <div class=sectbody> <div class=sectline>N:= 10^100: # to get all terms <= N</div> <div class=sectline>A:= {3}:</div> <div class=sectline>for n from 2 while 2*n^n < N do</div> <div class=sectline> for k from n+1 do if igcd(n, k)=1 then</div> <div class=sectline> a:= n^k + k^n;</div> <div class=sectline> if a > N then break fi;</div> <div class=sectline> if isprime(a) then A:= A union {a} fi fi;</div> <div class=sectline> od</div> <div class=sectline>od:</div> <div class=sectline>A; # if using Maple 11 or earlier, uncomment the next line</div> <div class=sectline># sort(convert(A, list)); # <a href="/wiki/User:Robert_Israel">Robert Israel</a>, Apr 13 2015</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>a = {3}; Do[Do[k = m^n + n^m; If[PrimeQ[k], AppendTo[a, k]], {m, 2, n}], {n, 2, 100}]; Union[a] (* <a href="/wiki/User:Artur_Jasinski">Artur Jasinski</a> *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(PARI) f(x)=my(L=log(x)); L/lambertw(L) \\ finds y such that y^y == x</div> <div class=sectline>list(lim)=my(v=List()); for(x=2, f(lim/2), my(y=x+1, t); while((t=x^y+y^x)<=lim, if(ispseudoprime(t), listput(v, t)); y+=2)); Set(v) \\ <a href="/wiki/User:Charles_R_Greathouse_IV">Charles R Greathouse IV</a>, Oct 28 2014</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A061119" title="Primes in the sequence n^2 + 2^n (A001580).">A061119</a> (primes where one of x,y is 2), <a href="/A064539" title="Numbers n such that 2^n + n^2 is prime.">A064539</a> (non-2 values where one of x,y is 2), <a href="/A253471" title="Numbers k such that 3^k + k^3 is prime.">A253471</a> (non-3 values where one of x,y is 3), <a href="/A073499" title="Numbers n such that n^(n+1) + (n+1)^n is prime.">A073499</a> (subset listing y where x = y+1), <a href="/A076980" title="Leyland numbers: 3, together with numbers expressible as n^k + k^n nontrivially, i.e., n,k > 1 (to avoid n = (n-1)^1 + 1^(n-1)).">A076980</a> (Leyland numbers).</div> <div class=sectline>Sequence in context: <a href="/A292082" title="Primes p such that (p^2 - 1) / 2 is a square (A000290).">A292082</a> <a href="/A001601" title="a(n) = 2*a(n-1)^2 - 1, if n>1. a(0)=1, a(1)=3.">A001601</a> <a href="/A061119" title="Primes in the sequence n^2 + 2^n (A001580).">A061119</a> * <a href="/A049985" title="Primes of form n! - (n-1)! - 1.">A049985</a> <a href="/A126579" title="a(n) = numerator of r_n, where r_0 =0, r_1 =1, r_{n+1} = the continued fraction (of rational terms) [r_0,r_1,r_2,r_3,...,r_n].">A126579</a> <a href="/A309060" title="Least k such that the rank of the elliptic curve y^2 = x^3 + k^2*x is n.">A309060</a></div> <div class=sectline>Adjacent sequences: <a href="/A094130" title="Decimal expansion of (e/Pi)^(1/4).">A094130</a> <a href="/A094131" title="Decimal expansion of (e/Pi)^(1/6).">A094131</a> <a href="/A094132" title="Decimal expansion of (e/Pi)^(1/7).">A094132</a> * <a href="/A094134" title="Decimal expansion of (e/Pi)^(1/5).">A094134</a> <a href="/A094135" title="Decimal expansion of (e/Pi)^(1/8).">A094135</a> <a href="/A094136" title="Values x of smallest positive pair (x,y) satisfying x^2 - 2*y^2 = -+d, where d=A058529(n).">A094136</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></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:Lekraj_Beedassy">Lekraj Beedassy</a>, May 04 2004</div> </div> </div> <div class=section> <div class=sectname>EXTENSIONS</div> <div class=sectbody> <div class=sectline>Corrected and extended by <a href="/wiki/User:Jens_Kruse_Andersen">Jens Kruse Andersen</a>, Oct 26 2007</div> <div class=sectline>Edited by <a href="/wiki/User:Hans_Havermann">Hans Havermann</a>, Apr 10 2015</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 16:05 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>