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<!doctype html> <html lang="en" class="numbertheory"> <head> <title>Self Number -- from Wolfram MathWorld</title> <meta name="DC.Title" content="Self Number" /> <meta name="DC.Creator" content="Weisstein, Eric W." /> <meta name="DC.Description" content="A number (usually base 10 unless specified otherwise) which has no digitaddition generator. Such numbers were originally called Colombian numbers (S. 1974). There are infinitely many such numbers, since an infinite sequence of self numbers can be generated from the recurrence relation C_k=8·10^(k-1)+C_(k-1)+8, (1) for k=2, 3, ..., where C_1=9. The first few self numbers are 1, 3, 5, 7, 9, 20, 31, 42, 53, 64, 75, 86, 97, ... (OEIS A003052). An infinite number of 2-self numbers (i.e.,..." /> <meta name="description" content="A number (usually base 10 unless specified otherwise) which has no digitaddition generator. Such numbers were originally called Colombian numbers (S. 1974). There are infinitely many such numbers, since an infinite sequence of self numbers can be generated from the recurrence relation C_k=8·10^(k-1)+C_(k-1)+8, (1) for k=2, 3, ..., where C_1=9. The first few self numbers are 1, 3, 5, 7, 9, 20, 31, 42, 53, 64, 75, 86, 97, ... (OEIS A003052). An infinite number of 2-self numbers (i.e.,..." /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:Number Theory:Special Numbers:Digit-Related Numbers" /> <meta name="DC.Rights" content="Copyright 1999-2024 Wolfram Research, Inc. 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Such numbers were originally called Colombian numbers (S. 1974). There are infinitely many such numbers, since an infinite sequence of self numbers can be generated from the recurrence relation C_k=8·10^(k-1)+C_(k-1)+8, (1) for k=2, 3, ..., where C_1=9. The first few self numbers are 1, 3, 5, 7, 9, 20, 31, 42, 53, 64, 75, 86, 97, ... (OEIS A003052). An infinite number of 2-self numbers (i.e.,..."> <meta name="twitter:card" content="summary_large_image"> <meta name="twitter:site" content="@WolframResearch"> <meta name="twitter:title" content="Self Number -- from Wolfram MathWorld"> <meta name="twitter:description" content="A number (usually base 10 unless specified otherwise) which has no digitaddition generator. Such numbers were originally called Colombian numbers (S. 1974). There are infinitely many such numbers, since an infinite sequence of self numbers can be generated from the recurrence relation C_k=8·10^(k-1)+C_(k-1)+8, (1) for k=2, 3, ..., where C_1=9. The first few self numbers are 1, 3, 5, 7, 9, 20, 31, 42, 53, 64, 75, 86, 97, ... (OEIS A003052). 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method="get" action="/search/" name="search" id="search-mobile" accept-charset="UTF-8"> <input type="text" name="query" placeholder="Search" id="searchFieldMobile"> <img src="/images/header/search-icon.png" width="18" height="18" id="mobile-search" class="search-btn" title="Search" alt="Search" onclick="submitForm()"> </form> <nav class="topics-nav"> <a href="/topics/Algebra.html" id="sidebar-algebra"> Algebra </a> <a href="/topics/AppliedMathematics.html" id="sidebar-appliedmathematics"> Applied Mathematics </a> <a href="/topics/CalculusandAnalysis.html" id="sidebar-calculusandanalysis"> Calculus and Analysis </a> <a href="/topics/DiscreteMathematics.html" id="sidebar-discretemathematics"> Discrete Mathematics </a> <a href="/topics/FoundationsofMathematics.html" id="sidebar-foundationsofmathematics"> Foundations of Mathematics </a> <a href="/topics/Geometry.html" id="sidebar-geometry"> Geometry </a> <a href="/topics/HistoryandTerminology.html" id="sidebar-historyandterminology"> History and Terminology </a> <a href="/topics/NumberTheory.html" id="sidebar-numbertheory"> Number Theory </a> <a href="/topics/ProbabilityandStatistics.html" id="sidebar-probabilityandstatistics"> Probability and Statistics </a> <a href="/topics/RecreationalMathematics.html" id="sidebar-recreationalmathematics"> Recreational Mathematics </a> <a href="/topics/Topology.html" id="sidebar-topology"> Topology </a> </nav> <nav class="secondary-nav"> <a href="/letters/"> Alphabetical Index </a> <a href="/whatsnew/"> New in MathWorld </a> </nav> </section> <section id="content">  <nav class="breadcrumbs"><ul class="breadcrumb"> <li> <a href="/topics/NumberTheory.html">Number Theory</a> </li> <li> <a href="/topics/SpecialNumbers.html">Special Numbers</a> </li> <li> <a href="/topics/Digit-RelatedNumbers.html">Digit-Related Numbers</a> </li> </ul></nav>   <h1>Self Number</h1>  <hr class="margin-t-1-8 margin-b-3-4">   <div class="entry-content"> <p> A number (usually base 10 unless specified otherwise) which has no <a href="/DigitadditionGenerator.html">digitaddition generator</a>. Such numbers were originally called Colombian numbers (S. 1974). There are infinitely many such numbers, since an infinite sequence of self numbers can be generated from the <a href="/RecurrenceRelation.html">recurrence relation</a> </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/SelfNumber/NumberedEquation1.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="171" height="21" alt=" C_k=8·10^(k-1)+C_(k-1)+8, " /></td><td align="right" width="3"> <div id="eqn1" class="eqnum"> (1) </div> </td></tr> </table> </div> <p> for <img src="/images/equations/SelfNumber/Inline1.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="39" height="21" alt="k=2" />, 3, ..., where <img src="/images/equations/SelfNumber/Inline2.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="49" height="21" alt="C_1=9" />. The first few self numbers are 1, 3, 5, 7, 9, 20, 31, 42, 53, 64, 75, 86, 97, ... (OEIS <a href="http://oeis.org/A003052">A003052</a>). </p> <p> An infinite number of 2-self numbers (i.e., base-2 self numbers) can be generated by the sequence </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/SelfNumber/NumberedEquation2.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="126" height="20" alt=" C_k=2^j+C_(k-1)+1 " /></td><td align="right" width="3"> <div id="eqn2" class="eqnum"> (2) </div> </td></tr> </table> </div> <p> for <img src="/images/equations/SelfNumber/Inline3.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="39" height="21" alt="k=1" />, 2, ..., where <img src="/images/equations/SelfNumber/Inline4.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="49" height="21" alt="C_1=1" /> and <img src="/images/equations/SelfNumber/Inline5.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="8" height="21" alt="j" /> is the number of digits in <img src="/images/equations/SelfNumber/Inline6.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="33" height="22" alt="C_(k-1)" />. An infinite number of <img src="/images/equations/SelfNumber/Inline7.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" />-self numbers can be generated from the sequence </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/SelfNumber/NumberedEquation3.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="222" height="21" alt=" C_k=(n-2)n^(k-1)+C_(k-1)+(n-2) " /></td><td align="right" width="3"> <div id="eqn3" class="eqnum"> (3) </div> </td></tr> </table> </div> <p> for <img src="/images/equations/SelfNumber/Inline8.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="39" height="21" alt="k=2" />, 3, ..., and </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/SelfNumber/NumberedEquation4.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="159" height="53" alt=" C_1={n-1 for n even; n-2 for n odd. " /></td><td align="right" width="3"> <div id="eqn4" class="eqnum"> (4) </div> </td></tr> </table> </div> <p> Joshi (1973) proved that if <img src="/images/equations/SelfNumber/Inline9.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="k" /> is <a href="/OddNumber.html">odd</a>, then <img src="/images/equations/SelfNumber/Inline10.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="m" /> is a <img src="/images/equations/SelfNumber/Inline11.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="k" />-self number <a href="/Iff.html">iff</a> <img src="/images/equations/SelfNumber/Inline12.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="m" /> is <a href="/OddNumber.html">odd</a>. Patel (1991) proved that <img src="/images/equations/SelfNumber/Inline13.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="22" height="21" alt="2k" />, <img src="/images/equations/SelfNumber/Inline14.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="50" height="21" alt="4k+2" />, and <img src="/images/equations/SelfNumber/Inline15.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="85" height="21" alt="k^2+2k+1" /> are <img src="/images/equations/SelfNumber/Inline16.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="k" />-self numbers in every <a href="/EvenNumber.html">even</a> base <img src="/images/equations/SelfNumber/Inline17.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="37" height="21" alt="k>=4" />. </p> </div>  <hr class="margin-b-1-1-4"> <div class="c-777 entry-secondary-content">  <h2>See also</h2><a href="/Digitaddition.html">Digitaddition</a>       <h2>Explore with Wolfram|Alpha</h2> <div id="WAwidget"> <div class="WAwidget-wrapper"> <img alt="WolframAlpha" title="WolframAlpha" src="/images/wolframalpha/WA-logo.png" width="136" height="20"> <form name="wolframalpha" action="https://www.wolframalpha.com/input/" target="_blank"> <input type="text" name="i" class="search" placeholder="Solve your math problems and get step-by-step solutions" value=""> <button type="submit" title="Evaluate on WolframAlpha"></button> </form> </div> <div class="WAwidget-wrapper try"> <p class="text-align-r"> More things to try: </p> <ul> <li> <a target="_blank" href="http://www.wolframalpha.com/input/?i=alladi-Grinstead+constant"> alladi-Grinstead constant </a> </li> <li><a target="_blank" href="https://www.wolframalpha.com/input/?i=Busy+Beaver+4-state+2-color">Busy Beaver 4-state 2-color</a></li> <li><a target="_blank" href="https://www.wolframalpha.com/input/?i=fourier+mellin+integral">fourier mellin integral</a></li> </ul> </div> </div>   <h2>References</h2><cite>Cai, T. "On <img src="/images/equations/SelfNumber/Inline18.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="k" />-Self Numbers and Universal Generated Numbers." <i>Fib. Quart.</i> <b>34</b>, 144-146, 1996.</cite><cite>Gardner, M. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0716719258/ref=nosim/ericstreasuretro">Time Travel and Other Mathematical Bewilderments.</a></i> New York: W. H. Freeman, pp. 115-117, 122, 1988.</cite><cite>Joshi, V. S. Ph.D. dissertation. Gujarat University, Ahmadabad, 1973.</cite><cite>Kaprekar, D. R. <i>The Mathematics of New Self-Numbers.</i> Devaiali, pp. 19-20, 1963.</cite><cite>Patel, R. B. "Some Tests for <img src="/images/equations/SelfNumber/Inline19.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="k" />-Self Numbers." <i>Math. Student</i> <b>56</b>, 206-210, 1991.</cite><cite>S., B. R. "Solution to Problem E 2048." <i>Amer. Math. Monthly</i> <b>81</b>, 407, 1974.</cite><cite>Sloane, N. J. A. Sequence <a href="http://oeis.org/A003052">A003052</a>/M2404 in "The On-Line Encyclopedia of Integer Sequences."</cite><h2>Referenced on Wolfram|Alpha</h2><a href="http://www.wolframalpha.com/entities/mathworld/self_number/1t/4c/ae/" title="Self Number" target="_blank">Self Number</a>   <h2>Cite this as:</h2> <p> <a href="/about/author.html">Weisstein, Eric W.</a> "Self Number." 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