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<!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>Demo1</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/demo1.html" 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=%2fdemo1.html">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="Demo1"></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> <center><h2>Demonstration of the</h2> <a href="http://oeis.org"><h2>On-Line Encyclopedia of Integer Sequences® (OEIS®)</h2></a> <h2>(Start)</h2></center><p> <center> <img src="bline.gif" alt=" "> </center> <p> <p> <ul> <li>This sequence of pages will show some of the ways that the <a href="http://oeis.org"><strong>On-Line Encyclopedia of Integer Sequences</strong></a> (or <a href="http://oeis.org"><strong>OEIS</strong></a>) can be used. </li> <p> <li>Let's begin right away with an example of a beautiful sequence from the database, sequence <a href="http://oeis.org/A000037">A000037</a>.<br> This entry illustrates most of the features that you will see in a typical OEIS entry.<br> (The following is a snapshot of the entry as it was on Sep 27 2011 — click the A-number to see the current version. The same thing applies to all the snapshots of sequences shown in these demonstration pages.) <p> <center> <table width="750" width="100%" cellspacing="0" cellpadding="0" border="0"> <tr height="1"><td> <tr height=1 bgcolor="#76767F"><td> <tr bgcolor="#EEEEFF"><td valign=top> <table width="100%" cellspacing="0" cellpadding="0" border="0"> <tr> <td valign=top align=left width=100> <a href="http://oeis.org/A000037">A000037</a> <td width=5> <td valign=top align=left> Numbers that are not squares. <br><font size=-1>(Formerly M0613 N0223)</font> <td width=2> <td valign=top align=right> <font size=-2>56</font> </table> <tr><td valign=top> <table cellspacing="0" cellpadding="2" cellborder="0"> <tr> <td width="20"> <td width="710"> <tt>2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99</tt> <font size=-1>(<a href="/A000037/list">list</a>; <a href="/A000037/graph">graph</a>; <a href="/search?q=A000037+-id:A000037">refs</a>; <a href="/A000037/listen">listen</a>; <a href="/history?seq=A000037">history</a>; <a href="/edit?seq=A000037">edit</a>; <a href="/A000037/internal">internal format</a>)</font> </table> <tr><td valign=top> <table cellspacing="0" cellpadding="2" cellborder="0"> <tr> <td width="20"> <td valign=top align=left width=100> <font size=-2>OFFSET</font> <td width=600> <p style="text-indent: -1em; margin-left: 1em; margin-top: 0; margin-bottom: 0;"><tt>1,1</tt> <tr> <td width="20"> <td valign=top align=left width=100> <font size=-2>COMMENTS</font> <td width=600> <p style="text-indent: -1em; margin-left: 1em; margin-top: 0; margin-bottom: 0;"><tt>Note the remarkable formula for the n-th term (see the FORMULA section)!</tt> <p style="text-indent: -1em; margin-left: 1em; margin-top: 0; margin-bottom: 0;"><tt>These are the natural numbers with an even number of divisors. The number of divisors is odd for the complementary sequence, the squares (sequence <a href="/A000290" title="The squares: a(n) = n^2.">A000290</a>) and the numbers for which the number of divisors is divisible by 3 is sequence <a href="/A059269" title="Number of divisors d(n) is divisible by 3.">A059269</a>. - Ola Veshta (olaveshta(AT)my-deja.com), Apr 04 2001</tt> <p style="text-indent: -1em; margin-left: 1em; margin-top: 0; margin-bottom: 0;"><tt>Also, a(n) = largest integer m not equal to n such that n = (floor(n^2/m) + m)/2. - Alexander R. Povolotsky (pevnev(AT)juno.com), Feb 10 2008</tt> <p style="text-indent: -1em; margin-left: 1em; margin-top: 0; margin-bottom: 0;"><tt><a href="/A010052" title="Characteristic function of squares: 1 if n is a square else 0.">A010052</a>(a(n)) = 0. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jan 26 2010]</tt> <p style="text-indent: -1em; margin-left: 1em; margin-top: 0; margin-bottom: 0;"><tt><a href="/A173517" title="a(n) = 0 unless n is a nonsquare in which case a(n) = k.">A173517</a>(a(n)) = n; a(n)^2 = <a href="/A030140" title="The nonsquares squared.">A030140</a>(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 20 2010]</tt> <p style="text-indent: -1em; margin-left: 1em; margin-top: 0; margin-bottom: 0;"><tt>Special case of formula from Farhi for positive integers which are not r-th powers [Jonathan Vos Post, May 5, 2011].</tt> <p style="text-indent: -1em; margin-left: 1em; margin-top: 0; margin-bottom: 0;"><tt>Union of <a href="/A007969" title="Rectangular numbers.">A007969</a> and <a href="/A007970" title="Rhombic numbers.">A007970</a>; <a href="/A007968" title="Type of happy factorization of n.">A007968</a>(a(n)) > 0. [Reinhard Zumkeller, Jun 18 2011]</tt> <tr> <td width="20"> <td valign=top align=left width=100> <font size=-2>REFERENCES</font> <td width=600> <p style="text-indent: -1em; margin-left: 1em; margin-top: 0; margin-bottom: 0;"><tt>A. J. dos Reis and D. M. Silberger, "Generating nonpowers by formula", Mathematics Magazine 63 (1990), pp. 53-55.</tt> <p style="text-indent: -1em; margin-left: 1em; margin-top: 0; margin-bottom: 0;"><tt>J. Lambek and L. Moser, "Inverse and complementary sequences of natural numbers", The American Mathematical Monthly, Vol. 61, No. 7 (1954), 454-458, doi 10.2307/2308078, see example 4 (includes the formula). [From Nicolas Normand (Nicolas.Normand(AT)polytech.univ-nantes.fr), Nov 24 2009]</tt> <p style="text-indent: -1em; margin-left: 1em; margin-top: 0; margin-bottom: 0;"><tt>M. A. Nyblom, "Some curious sequences involving floor and ceiling functions", American Mathematical Monthly 109 (2002), pp. 559-564.</tt> <p style="text-indent: -1em; margin-left: 1em; margin-top: 0; margin-bottom: 0;"><tt>N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).</tt> <p style="text-indent: -1em; margin-left: 1em; margin-top: 0; margin-bottom: 0;"><tt>N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).</tt> <tr> <td width="20"> <td valign=top align=left width=100> <font size=-2>LINKS</font> <td width=600> <p style="text-indent: -1em; margin-left: 1em; margin-top: 0; margin-bottom: 0;"><tt>N. J. A. Sloane, <a href="/A000037/b000037.txt">Table of n, a(n) for n = 1..10000</a></tt> <p style="text-indent: -1em; margin-left: 1em; margin-top: 0; margin-bottom: 0;"><tt>Bakir Farhi, <a href="http://arxiv.org/abs/1105.1127">An explicit formula generating the non-Fibonacci numbers</a>, May 5, 2011.</tt> <p style="text-indent: -1em; margin-left: 1em; margin-top: 0; margin-bottom: 0;"><tt>S. R. Finch, <a href="http://algo.inria.fr/bsolve/">Class number theory</a></tt> <p style="text-indent: -1em; margin-left: 1em; margin-top: 0; margin-bottom: 0;"><tt>Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SquareNumber.html">Square Number</a></tt> <p style="text-indent: -1em; margin-left: 1em; margin-top: 0; margin-bottom: 0;"><tt>Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ContinuedFraction.html">Continued Fraction</a></tt> <tr> <td width="20"> <td valign=top align=left width=100> <font size=-2>FORMULA</font> <td width=600> <p style="text-indent: -1em; margin-left: 1em; margin-top: 0; margin-bottom: 0;"><tt>a(n) = n + [1/2 + sqrt(n)].</tt> <p style="text-indent: -1em; margin-left: 1em; margin-top: 0; margin-bottom: 0;"><tt>Another formula: a(n) = n + [ sqrt( n + [ sqrt n ] ) ].</tt> <p style="text-indent: -1em; margin-left: 1em; margin-top: 0; margin-bottom: 0;"><tt>a(n) = <a href="/A000194" title="n appears 2n times; also nearest integer to square root of n.">A000194</a>(n) + n = floor(1/2 *(1 + sqrt(4*n-3)))+ n. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jun 14 2009]</tt> <p style="text-indent: -1em; margin-left: 1em; margin-top: 0; margin-bottom: 0;"><tt>d(a(n))=even. [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Oct 20 2009]</tt> <p style="text-indent: -1em; margin-left: 1em; margin-top: 0; margin-bottom: 0;"><tt>a(n) = <a href="/A000194" title="n appears 2n times; also nearest integer to square root of n.">A000194</a>(n) + n.</tt> <tr> <td width="20"> <td valign=top align=left width=100> <font size=-2>EXAMPLE</font> <td width=600> <p style="text-indent: -1em; margin-left: 1em; margin-top: 0; margin-bottom: 0;"><tt>For example note that the squares 1, 4, 9, 16 are not included.</tt> <p style="text-indent: -1em; margin-left: 1em; margin-top: 0; margin-bottom: 0;"><tt>a(<a href="/A002061" title="Central polygonal numbers: n^2 - n + 1.">A002061</a>(n)) = a(n^2-n+1) = <a href="/A002522" title="n^2 + 1.">A002522</a>(n) = n^2 + 1. <a href="/A002061" title="Central polygonal numbers: n^2 - n + 1.">A002061</a>(n) = central polygonal numbers (n^2-n+1). <a href="/A002522" title="n^2 + 1.">A002522</a>(n) = numbers of the form n^2 + 1. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jun 21 2009]</tt> <tr> <td width="20"> <td valign=top align=left width=100> <font size=-2>MAPLE</font> <td width=600> <p style="text-indent: -1em; margin-left: 1em; margin-top: 0; margin-bottom: 0;"><tt><a href="/A000037" title="Numbers that are not squares.">A000037</a> := n->n+floor(1/2+sqrt(n));</tt> <tr> <td width="20"> <td valign=top align=left width=100> <font size=-2>MATHEMATICA</font> <td width=600> <p style="text-indent: -1em; margin-left: 1em; margin-top: 0; margin-bottom: 0;"><tt>f[n_] := (n + Floor[Sqrt[n + Floor[Sqrt[n]]]]); Table[ f[n], {n, 71}] (from Robert G. Wilson v Sep 24 2004)</tt> <p style="text-indent: -1em; margin-left: 1em; margin-top: 0; margin-bottom: 0;"><tt>f[n_]:=Round[Sqrt[n]]; lst={}; Do[AppendTo[lst, n+f[n]], {n, 0, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 13 2009]</tt> <tr> <td width="20"> <td valign=top align=left width=100> <font size=-2>PROG</font> <td width=600> <p style="text-indent: -1em; margin-left: 1em; margin-top: 0; margin-bottom: 0;"><tt>(MAGMA) [n : n in [1..1000] | not IsSquare(n) ];</tt> <p style="text-indent: -1em; margin-left: 1em; margin-top: 0; margin-bottom: 0;"><tt>(MAGMA) at:=0; for n in [1..10000] do if not IsSquare(n) then at:=at+1; print at, n; end if; end for;</tt> <p style="text-indent: -1em; margin-left: 1em; margin-top: 0; margin-bottom: 0;"><tt>(PARI) a(n)=if(n<0, 0, n+(1+sqrtint(4*n))\2)</tt> <tr> <td width="20"> <td valign=top align=left width=100> <font size=-2>CROSSREFS</font> <td width=600> <p style="text-indent: -1em; margin-left: 1em; margin-top: 0; margin-bottom: 0;"><tt>Cf. <a href="/A007412" title="The noncubes: n + [ (n + [ n^{1/3} ])^{1/3} ].">A007412</a>, <a href="/A000005" title="d(n) (also called tau(n) or sigma_0(n)), the number of divisors of n.">A000005</a>, <a href="/A000290" title="The squares: a(n) = n^2.">A000290</a>, <a href="/A059269" title="Number of divisors d(n) is divisible by 3.">A059269</a>.</tt> <p style="text-indent: -1em; margin-left: 1em; margin-top: 0; margin-bottom: 0;"><tt>Cf. <a href="/A134986" title="a(n) = smallest integer m not equal to n such that n = (floor(n^2/m) + m)/2.">A134986</a>.</tt> <p style="text-indent: -1em; margin-left: 1em; margin-top: 0; margin-bottom: 0;"><tt>Cf. <a href="/A087153" title="Number of partitions of n into nonsquares.">A087153</a>, <a href="/A172151" title="Number of partitions of n into two nonsquares.">A172151</a>. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jan 26 2010]</tt> <p style="text-indent: -1em; margin-left: 1em; margin-top: 0; margin-bottom: 0;"><tt>Sequence in context: <a href="/A072099" title="First term = 1 and after, if n appears, n*(rank of n) does not.">A072099</a> <a href="/A046841" title="Sum of divisors divides sum of cubes of divisors.">A046841</a> <a href="/A164514" title="1 followed by numbers that are not squares.">A164514</a> * <a href="/A028761" title="Nonsquares mod 48.">A028761</a> <a href="/A028809" title="Nonsquares mod 96.">A028809</a> <a href="/A028785" title="Nonsquares mod 72.">A028785</a></tt> <p style="text-indent: -1em; margin-left: 1em; margin-top: 0; margin-bottom: 0;"><tt>Adjacent sequences: <a href="/A000034" title="A simple periodic sequence.">A000034</a> <a href="/A000035" title="A simple periodic sequence.">A000035</a> <a href="/A000036" title="Let A(n) = #{(i,j): i^2 + j^2 <= n}, V(n) = Pi*n, P(n) = A(n) - V(n); A000099 gives values of n where |P(n)| sets a new reco...">A000036</a> * <a href="/A000038" title="Twice A000007.">A000038</a> <a href="/A000039" title="Coefficient of q^(2n) in the series expansion of Ramanujan's mock theta function f(q).">A000039</a> <a href="/A000040" title="The prime numbers.">A000040</a></tt> <tr> <td width="20"> <td valign=top align=left width=100> <font size=-2>KEYWORD</font> <td width=600> <p style="text-indent: -1em; margin-left: 1em; margin-top: 0; margin-bottom: 0;"><tt><span title="it is very easy to produce terms of sequence">easy</span>,<span title="a sequence of nonnegative numbers">nonn</span>,<span title="an exceptionally nice sequence">nice</span>,<span title="edited within the last two weeks">changed</span></tt> <tr> <td width="20"> <td valign=top align=left width=100> <font size=-2>AUTHOR</font> <td width=600> <p style="text-indent: -1em; margin-left: 1em; margin-top: 0; margin-bottom: 0;"><tt>N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)</tt> <tr> <td width="20"> <td valign=top align=left width=100> <font size=-2>EXTENSIONS</font> <td width=600> <p style="text-indent: -1em; margin-left: 1em; margin-top: 0; margin-bottom: 0;"><tt>Edited by Charles R Greathouse IV (charles.greathouse(AT)case.edu), Oct 30 2009</tt> </table> <tr height=10><td> </table> </center> Note the surprising formula for the n-th term! <p> </li> <center> <img src="bline.gif" alt=" "> </center> <p> <li> To follow the sequence of pages, click only the direction buttons <IMG SRC="StartShade.gif" width=20 height=20 border=0 alt=" "> <IMG SRC="PrevPageShade.gif" width=20 height=20 border=0 alt=" "> <a href="demo2.html"> <IMG SRC="NextPage.gif" width=20 height=20 border=0 alt=" "></a> <a href="demol.html"> <IMG SRC="LastPage.gif" width=20 height=20 border=0 alt=" "></a> at the bottom of the pages. (If you click on any of the active links you will have to use your browser's "Back" button to return here.) </li> <p> <li> Although these demonstration pages are in English, note that the <a href="http://oeis.org">main lookup page</a> has been translated into many other languages. </li> <p> <li> The main URL for the database is <a href="http://oeis.org"><strong>http://oeis.org</strong></a>. </li> <p> <li> The sequence of demonstration pages contains the following pages: <p> <ul> <li><a href="demo1.html">Starting page (this page)</a></li> <li><a href="demo2.html">Identifying a sequence - description of database</a></li> <li><a href="demo3.html">Identifying a sequence: supplying a formula</a></li> <li><a href="demo4.html">Identifying a sequence: a puzzle</a></li> <li><a href="demo5.html">Identifying a sequence: a sequence from a chemical journal</a></li> <li><a href="demo6.html">Finding latest information about a sequence</a></li> <li><a href="demo7.html">What are the Bell numbers?</a></li> <li><a href="demo8.html">A binomial coefficient sum</a></li> <li><a href="demob.html">Browsing and the WebCam</a></li> <li><a href="demoe.html">The email server</a></li> <li><a href="demos.html">Superseeker</a></li> <li><a href="demor.html">Fractions, arrays, real numbers, etc.</a></li> <li><a href="demok.html">The book versions</a></li> <li><a href="demol.html">Comments from users</a></li> </ul> </li> <p> <li> Two other pages worth looking at are: <p> <ul> <li><a href="http://oeis.org/wiki/Welcome">Welcome to the OEIS</a></li> <li><a href="http://oeis.org/wiki/Works_Citing_OEIS">List of works citing the OEIS</a></li> </ul> </li> <p> </ul> <p> <div align="center"> <img src="bline.gif" alt=" "> <p> Click the single right arrow to go to the next page. <p> <a href="demo2.html"> <IMG SRC="NextPage.gif" alt="NEXT!"></a> <a href="demol.html"> <IMG SRC="LastPage.gif" alt="Last"></a> <a href="index.html"> <IMG SRC="arrow4upwte.gif" alt="Main page"></a> <p> </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 01:23 EST 2024. 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