CINXE.COM
A005529 - 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>A005529 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A005529" 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=%2fA005529">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="A005529 - 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> A005529 </div> <div class=seqname> Primitive prime factors of the sequence k^2 + 1 (<a href="/A002522" title="a(n) = n^2 + 1.">A002522</a>) in the order that they are found. <br><font size=-1>(Formerly M1505)</font> </div> </div> <div class=scorerefs> 10 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>2, 5, 17, 13, 37, 41, 101, 61, 29, 197, 113, 257, 181, 401, 97, 53, 577, 313, 677, 73, 157, 421, 109, 89, 613, 1297, 137, 761, 1601, 353, 149, 1013, 461, 1201, 1301, 541, 281, 2917, 3137, 673, 1741, 277, 1861, 769, 397, 241, 2113, 4357, 449, 2381, 2521, 5477</div> <div class=seqdatalinks> (<a href="/A005529/list">list</a>; <a href="/A005529/graph">graph</a>; <a href="/search?q=A005529+-id:A005529">refs</a>; <a href="/A005529/listen">listen</a>; <a href="/history?seq=A005529">history</a>; <a href="/search?q=id:A005529&fmt=text">text</a>; <a href="/A005529/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>Primes associated with Stormer numbers.</div> <div class=sectline>See <a href="/A002313" title="Primes congruent to 1 or 2 modulo 4; or, primes of form x^2 + y^2; or, -1 is a square mod p.">A002313</a> for the sorted list of primes. It can be shown that k^2 + 1 has at most one primitive prime factor; the other prime factors divide m^2 + 1 for some m < k. When k^2 + 1 has a primitive prime factor, k is a Stormer number (<a href="/A005528" title="St酶rmer numbers or arc-cotangent irreducible numbers: numbers k such that the largest prime factor of k^2 + 1 is >= 2*k.">A005528</a>), otherwise a non-Stormer number (<a href="/A002312" title="Arc-cotangent reducible numbers or non-St酶rmer numbers: largest prime factor of k^2 + 1 is less than 2*k.">A002312</a>).</div> </div> </div> <div class=section> <div class=sectname>REFERENCES</div> <div class=sectbody> <div class=sectline>John H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, p. 246.</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>J. Todd, Table of Arctangents. National Bureau of Standards, Washington, DC, 1951, p. vi.</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>T. D. Noe, <a href="/A005529/b005529.txt">Table of n, a(n) for n = 1..1000</a></div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/StormerNumber.html">Stormer Number.</a></div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimitivePrimeFactor.html">Primitive Prime Factor</a></div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>prms={}; Do[f=First/@FactorInteger[k^2+1]; p=Complement[f, prms]; prms=Join[prms, p], {k, 100}]; prms</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(Magma) V:=[]; for n in [1..75] do p:=Max([ x[1]: x in Factorization(n^2+1) ]); if not p in V then Append(~V, p); end if; end for; V; - <a href="/wiki/User:Klaus_Brockhaus">Klaus Brockhaus</a>, Oct 29 2008</div> <div class=sectline>(PARI) do(n)=my(v=List(), g=1, m, t, f); for(k=1, n, m=k^2+1; t=gcd(m, g); while(t>1, m/=t; t=gcd(m, t)); f=factor(m)[, 1]; if(#f, listput(v, f[1]); g*=f[1])); Vec(v) \\ <a href="/wiki/User:Charles_R_Greathouse_IV">Charles R Greathouse IV</a>, Jun 11 2017</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A002312" title="Arc-cotangent reducible numbers or non-St酶rmer numbers: largest prime factor of k^2 + 1 is less than 2*k.">A002312</a>, <a href="/A002313" title="Primes congruent to 1 or 2 modulo 4; or, primes of form x^2 + y^2; or, -1 is a square mod p.">A002313</a> (primes of the form 4k+1), <a href="/A002522" title="a(n) = n^2 + 1.">A002522</a>, <a href="/A005528" title="St酶rmer numbers or arc-cotangent irreducible numbers: numbers k such that the largest prime factor of k^2 + 1 is >= 2*k.">A005528</a>.</div> <div class=sectline>Sequence in context: <a href="/A033835" title="Smallest number > 1 equal to sum of n-th powers of its base-3 digits, or 0 if no such number exists (written in base 10).">A033835</a> <a href="/A366609" title="Smallest prime dividing 4^n + 1.">A366609</a> <a href="/A178198" title="Smallest k such that 37^k mod k = n">A178198</a> * <a href="/A259255" title="With a(1) = 1, a(n) is the smallest number not already in the sequence such that -1 + Product_{i=1..n} a(i) is a square.">A259255</a> <a href="/A274903" title="Largest prime factor of 4^n + 1.">A274903</a> <a href="/A206029" title="a(n) = sum of numbers k <= sigma(n) such that k is not equal to sigma(d) for any divisor d of n where sigma = A000203.">A206029</a></div> <div class=sectline>Adjacent sequences: <a href="/A005526" title="Maximal number of rational points that a (smooth, geometrically irreducible) curve of genus 3 over the finite field GF(q) ca...">A005526</a> <a href="/A005527" title="Maximal number of rational points on a curve of genus n over GF(2).">A005527</a> <a href="/A005528" title="St酶rmer numbers or arc-cotangent irreducible numbers: numbers k such that the largest prime factor of k^2 + 1 is >= 2*k.">A005528</a> * <a href="/A005530" title="Number of Boolean functions of n variables from Post class F(8,inf); number of degenerate Boolean functions of n variables.">A005530</a> <a href="/A005531" title="Decimal expansion of fifth root of 2.">A005531</a> <a href="/A005532" title="Decimal expansion of fifth root of 3.">A005532</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="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>Edited by <a href="/wiki/User:T._D._Noe">T. D. Noe</a>, Oct 02 2003</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 19:03 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>