CINXE.COM
A005528 - 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>A005528 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A005528" 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=%2fA005528">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="A005528 - 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> A005528 </div> <div class=seqname> St酶rmer numbers or arc-cotangent irreducible numbers: numbers k such that the largest prime factor of k^2 + 1 is >= 2*k. <br><font size=-1>(Formerly M0950)</font> </div> </div> <div class=scorerefs> 10 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>1, 2, 4, 5, 6, 9, 10, 11, 12, 14, 15, 16, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 33, 34, 35, 36, 37, 39, 40, 42, 44, 45, 48, 49, 51, 52, 53, 54, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 71, 74, 77, 78, 79, 80, 81, 82, 84, 85, 86, 87, 88, 89, 90, 92, 94, 95, 96</div> <div class=seqdatalinks> (<a href="/A005528/list">list</a>; <a href="/A005528/graph">graph</a>; <a href="/search?q=A005528+-id:A005528">refs</a>; <a href="/A005528/listen">listen</a>; <a href="/history?seq=A005528">history</a>; <a href="/search?q=id:A005528&fmt=text">text</a>; <a href="/A005528/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>Also numbers k such that k^2 + 1 has a primitive divisor, hence (by Everest & Harman, Theorem 1.4) 1.1n < a(n) < 1.88n for large enough n. They conjecture that a(n) ~ cn where c = 1/log 2 = 1.4426.... - <a href="/wiki/User:Charles_R_Greathouse_IV">Charles R Greathouse IV</a>, Nov 15 2014</div> <div class=sectline>Named after the Norwegian mathematician and astrophysicist Carl St酶rmer (1874-1957). - <a href="/wiki/User:Amiram_Eldar">Amiram Eldar</a>, Jun 08 2021</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>Graham Everest and Glyn Harman, On primitive divisors of n^2 + b, in Number Theory and Polynomials (James McKee and Chris Smyth, ed.), London Mathematical Society 2008.</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>John Todd, Table of Arctangents, National Bureau of Standards, Washington, DC, 1951, p. 2.</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>Amiram Eldar, <a href="/A005528/b005528.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe)</div> <div class=sectline>Graham Everest and Glyn Harman, <a href="http://arxiv.org/abs/math/0701234">On primitive divisors of n^2 + b</a>, arXiv:math/0701234 [math.NT], 2007.</div> <div class=sectline>Carl St酶rmer, <a href="https://archive.org/details/archivformathema1918961897oslo/page/n121/mode/2up">Sur l'application de la th茅orie des nombres entiers complexes 脿 la solution en nombres rationnels x_1 x_2... x_n c_1 c_2... c_n, k de l'茅quation: c_1 arc tg x_1 + c_2 arc tg x_2 + ... + c_n arc tg x_n = k * Pi/4</a>, Archiv for mathematik og naturvidenskab, Vol. 19, No. 3 (1896), pp. 1-96.</div> <div class=sectline>John Todd, <a href="http://www.jstor.org/stable/2305526">A problem on arc tangent relations</a>, Amer. Math. Monthly, 56 (1949), 517-528.</div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StormerNumber.html">St酶rmer Number</a>.</div> <div class=sectline>Wikipedia, <a href="https://en.wikipedia.org/wiki/St%C3%B8rmer_number">St酶rmer number</a>.</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>Select[Range[96], FactorInteger[#^2 + 1][[-1, 1]] >= 2 # &] (* <a href="/wiki/User:Jean-Fran莽ois_Alcover">Jean-Fran莽ois Alcover</a>, Apr 11 2011 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(PARI) is(n)=my(f=factor(n^2+1)[, 1]); f[#f]>=2*n \\ <a href="/wiki/User:Charles_R_Greathouse_IV">Charles R Greathouse IV</a>, Nov 14 2014</div> <div class=sectline>(Haskell)</div> <div class=sectline>a005528 n = a005528_list !! (n-1)</div> <div class=sectline>a005528_list = filter (\x -> 2 * x <= a006530 (x ^ 2 + 1)) [1..]</div> <div class=sectline>-- <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Jun 12 2015</div> <div class=sectline>(Python)</div> <div class=sectline>from sympy import factorint</div> <div class=sectline>def ok(n): return max(factorint(n*n + 1)) >= 2*n</div> <div class=sectline>print(list(filter(ok, range(1, 97)))) # <a href="/wiki/User:Michael_S._Branicky">Michael S. Branicky</a>, Aug 30 2021</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="/A006530" title="Gpf(n): greatest prime dividing n, for n >= 2; a(1)=1.">A006530</a>.</div> <div class=sectline>Cf. <a href="/A084925" title="Inverse hyperbolic cotangent irreducible numbers: positive integers such that the arccoth of these numbers form a basis for ...">A084925</a> (hyperbolic analog).</div> <div class=sectline>Sequence in context: <a href="/A143070" title="A positive integer n is included if the number of 0's in the binary representation of n is a power of 2 (including being pos...">A143070</a> <a href="/A340698" title="a(n) is the position of n^2 in the list of perfect powers.">A340698</a> <a href="/A206926" title="Numbers such that the number of contiguous palindromic bit patterns in their binary representation is minimal (for a given n...">A206926</a> * <a href="/A211030" title="Sum of all parts in the structure of the shell model of partitions of A135010 after n-th stage.">A211030</a> <a href="/A050015" title="Numbers k such that b(k) > b(k+1), where b=A050012.">A050015</a> <a href="/A153218" title="Numbers k such that 6k + 7 is prime.">A153218</a></div> <div class=sectline>Adjacent sequences: <a href="/A005525" title="Maximal number of rational points on a curve of genus 2 over GF(q), where q = A246655(n) is the n-th prime power > 1.">A005525</a> <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="/A005529" title="Primitive prime factors of the sequence k^2 + 1 (A002522) in the order that they are found.">A005529</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></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="an exceptionally nice sequence">nice</span>,<span title="it is very easy to produce terms of sequence">easy</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> and <a href="/wiki/User:J._H._Conway">J. H. Conway</a></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 November 24 16:42 EST 2024. Contains 378083 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>