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<!doctype html> <html lang="en" class="foundationsofmathematics numbertheory"> <head> <title>Smarandache Number -- from Wolfram MathWorld</title> <meta name="DC.Title" content="Smarandache Number" /> <meta name="DC.Creator" content="Weisstein, Eric W." /> <meta name="DC.Description" content="Consider the Consecutive Number Sequences formed by the concatenation of the first n positive integers: 1, 12, 123, 1234, ... (OEIS A007908; Smarandache 1993, Dumitrescu and Seleacu 1994, sequence 1; Mudge 1995; Stephan 1998; Wolfram 2002, p. 913). This sequence gives the digits of the Champernowne constant, and is sometimes also known as the Barbier infinite word (Allouche and Shallit 2003, pp. 114, 299, and 336). The terms up to n=9 are given by c_n = sum_(k=1)^(n)k·10^(n-k) (1) =..." /> <meta name="description" content="Consider the Consecutive Number Sequences formed by the concatenation of the first n positive integers: 1, 12, 123, 1234, ... (OEIS A007908; Smarandache 1993, Dumitrescu and Seleacu 1994, sequence 1; Mudge 1995; Stephan 1998; Wolfram 2002, p. 913). This sequence gives the digits of the Champernowne constant, and is sometimes also known as the Barbier infinite word (Allouche and Shallit 2003, pp. 114, 299, and 336). The terms up to n=9 are given by c_n = sum_(k=1)^(n)k·10^(n-k) (1) =..." /> <meta name="DC.Date.Created" scheme="W3CDTF" content="2015-11-04" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2016-12-05" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:Number Theory:Special Numbers:Digit-Related Numbers" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:Foundations of Mathematics:A New Kind of Science" /> <meta name="DC.Rights" content="Copyright 1999-2025 Wolfram Research, Inc. See https://mathworld.wolfram.com/about/terms.html for a full terms of use statement." /> <meta name="DC.Format" scheme="IMT" content="text/html" /> <meta name="DC.Identifier" scheme="URI" content="https://mathworld.wolfram.com/SmarandacheNumber.html" /> <meta name="DC.Language" scheme="RFC3066" content="en" /> <meta name="DC.Publisher" content="Wolfram Research, Inc." /> <meta name="DC.Relation.IsPartOf" scheme="URI" content="https://mathworld.wolfram.com/" /> <meta name="DC.Type" scheme="DCMIType" content="Text" /> <meta name="Last-Modified" content="2016-12-05" /> <meta property="og:image" content="https://mathworld.wolfram.com/images/socialmedia/share/ogimage_SmarandacheNumber.png"> <meta property="og:url" content="https://mathworld.wolfram.com/SmarandacheNumber.html"> <meta property="og:type" content="website"> <meta property="og:title" content="Smarandache Number -- from Wolfram MathWorld"> <meta property="og:description" content="Consider the Consecutive Number Sequences formed by the concatenation of the first n positive integers: 1, 12, 123, 1234, ... (OEIS A007908; Smarandache 1993, Dumitrescu and Seleacu 1994, sequence 1; Mudge 1995; Stephan 1998; Wolfram 2002, p. 913). This sequence gives the digits of the Champernowne constant, and is sometimes also known as the Barbier infinite word (Allouche and Shallit 2003, pp. 114, 299, and 336). The terms up to n=9 are given by c_n = sum_(k=1)^(n)k·10^(n-k) (1) =..."> <meta name="twitter:card" content="summary_large_image"> <meta name="twitter:site" content="@WolframResearch"> <meta name="twitter:title" content="Smarandache Number -- from Wolfram MathWorld"> <meta name="twitter:description" content="Consider the Consecutive Number Sequences formed by the concatenation of the first n positive integers: 1, 12, 123, 1234, ... (OEIS A007908; Smarandache 1993, Dumitrescu and Seleacu 1994, sequence 1; Mudge 1995; Stephan 1998; Wolfram 2002, p. 913). This sequence gives the digits of the Champernowne constant, and is sometimes also known as the Barbier infinite word (Allouche and Shallit 2003, pp. 114, 299, and 336). The terms up to n=9 are given by c_n = sum_(k=1)^(n)k·10^(n-k) (1) =..."> <meta name="twitter:image:src" content="https://mathworld.wolfram.com/images/socialmedia/share/ogimage_SmarandacheNumber.png"> <link rel="canonical" href="https://mathworld.wolfram.com/SmarandacheNumber.html" /> <meta http-equiv="x-ua-compatible" content="ie=edge"> <meta name="viewport" content="width=device-width, initial-scale=1"> <meta charset="utf-8"> <script async src="/common/javascript/analytics.js"></script> <script async src="//www.wolframcdn.com/consent/cookie-consent.js"></script> <script async src="/common/javascript/wal/latest/walLoad.js"></script> <link rel="stylesheet" href="/css/styles.css"> <link rel="preload" href="//www.wolframcdn.com/fonts/source-sans-pro/1.0/global.css" as="style" onload="this.onload=null;this.rel='stylesheet'"> <noscript><link rel="stylesheet" href="//www.wolframcdn.com/fonts/source-sans-pro/1.0/global.css"></noscript> </head> <body id="topics"> <main id="entry"> <div 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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><ul class="breadcrumb"> <li> <a href="/topics/FoundationsofMathematics.html">Foundations of Mathematics</a> </li> <li> <a href="/topics/ANewKindofScience.html">A New Kind of Science</a> </li> </ul></nav>   <h1>Smarandache Number</h1>  <hr class="margin-t-1-8 margin-b-3-4">  <div class="attachments text-align-r"> <a href="/notebooks/IntegerSequences/SmarandacheNumber.nb" download="SmarandacheNumber.nb"><img src="/images/entries/download-notebook-icon.png" width="26" height="27" alt="DOWNLOAD Mathematica Notebook" /><span>Download <span class="display-i display-n__600">Wolfram </span>Notebook</span></a> </div>  <div class="entry-content"> <p> Consider the <a href="/ConsecutiveNumberSequences.html">Consecutive Number Sequences</a> formed by the concatenation of the first <img src="/images/equations/SmarandacheNumber/Inline1.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" /> <a href="/PositiveInteger.html">positive integers</a>: 1, 12, 123, 1234, ... (OEIS <a href="http://oeis.org/A007908">A007908</a>; Smarandache 1993, Dumitrescu and Seleacu 1994, sequence 1; Mudge 1995; Stephan 1998; Wolfram 2002, p. <a href="http://www.wolframscience.com/nksonline/page-913-text">913</a>). This sequence gives the digits of the <a href="/ChampernowneConstant.html">Champernowne constant</a>, and is sometimes also known as the Barbier infinite word (Allouche and Shallit 2003, pp. 114, 299, and 336). The terms up to <img src="/images/equations/SmarandacheNumber/Inline2.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="39" height="21" alt="n=9" /> are given by </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/SmarandacheNumber/Inline3.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="15" height="20" alt="c_n" /></td><td align="center" width="14"><img src="/images/equations/SmarandacheNumber/Inline4.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/SmarandacheNumber/Inline5.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="79" height="51" alt="sum_(k=1)^(n)k·10^(n-k)" /></td><td align="right" width="10"> <div id="eqn1" class="eqnum"> (1) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/SmarandacheNumber/Inline6.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/SmarandacheNumber/Inline7.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/SmarandacheNumber/Inline8.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="142" height="26" alt="1/(81)(10^(n+1)-9n-10)." /></td><td align="right" width="10"> <div id="eqn2" class="eqnum"> (2) </div> </td></tr> </table> </div> <p> These are sometimes called Smarandache consecutive numbers, but in this work, the terms in the sequence will be called simply Smarandache numbers. Similarly, a Smarandache number that is prime will be called a <a href="/SmarandachePrime.html">Smarandache prime</a>. Surprisingly, no <a href="/SmarandachePrime.html">Smarandache primes</a> <img src="/images/equations/SmarandacheNumber/Inline9.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="47" height="21" alt="Sm(n)" /> exist for <img src="/images/equations/SmarandacheNumber/Inline10.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="85" height="21" alt="n<=344869" /> (Great Smarandache PRPrime search; Dec. 5, 2016). </p> <p> The number of digits of <img src="/images/equations/SmarandacheNumber/Inline11.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="47" height="21" alt="Sm(n)" /> can be computed by noticing the pattern in the following table, where </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/SmarandacheNumber/NumberedEquation1.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="114" height="22" alt=" d=|_log_(10)n_|+1 " /></td><td align="right" width="3"> <div id="eqn3" class="eqnum"> (3) </div> </td></tr> </table> </div> <p> is the number of digits in <img src="/images/equations/SmarandacheNumber/Inline12.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" />. </p> <div class="table-responsive"> <table align="center" class="mathworldtable"> <tr style=""><td align="left"><img src="/images/equations/SmarandacheNumber/Inline13.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="10" height="21" alt="d" /></td><td align="left"><img src="/images/equations/SmarandacheNumber/Inline14.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" /> range</td><td align="left">digits</td></tr><tr style=""><td align="left">1</td><td align="left">1-9</td><td align="left"><img src="/images/equations/SmarandacheNumber/Inline15.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" /></td></tr><tr style=""><td align="left">2</td><td align="left">10-99</td><td align="left"><img src="/images/equations/SmarandacheNumber/Inline16.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="88" height="21" alt="9+2(n-9)" /></td></tr><tr style=""><td align="left">3</td><td align="left">100-999</td><td align="left"><img src="/images/equations/SmarandacheNumber/Inline17.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="153" height="21" alt="9+90·2+3(n-99)" /></td></tr><tr style=""><td align="left">4</td><td align="left">1000-9999</td><td align="left"><img src="/images/equations/SmarandacheNumber/Inline18.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="227" height="21" alt="9+90·2+900·3+4(n-999)" /></td></tr> </table> </div> <p> By induction, the number of digits <img src="/images/equations/SmarandacheNumber/Inline19.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="37" height="21" alt="D(n)" /> in <img src="/images/equations/SmarandacheNumber/Inline20.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="47" height="21" alt="Sm(n)" /> can be written </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/SmarandacheNumber/Inline21.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="35" height="20" alt="D(n)" /></td><td align="center" width="14"><img src="/images/equations/SmarandacheNumber/Inline22.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/SmarandacheNumber/Inline23.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="217" height="54" alt="d(n+1-10^(d-1))+sum_(k=1)^(d-1)9k·10^(k-1)" /></td><td align="right" width="10"> <div id="eqn4" class="eqnum"> (4) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/SmarandacheNumber/Inline24.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/SmarandacheNumber/Inline25.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/SmarandacheNumber/Inline26.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="132" height="42" alt="(n+1)d-(10^d-1)/9," /></td><td align="right" width="10"> <div id="eqn5" class="eqnum"> (5) </div> </td></tr> </table> </div> <p> where the second term is the <a href="/Repunit.html">repunit</a> <img src="/images/equations/SmarandacheNumber/Inline27.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="19" height="21" alt="R_d" />. For <img src="/images/equations/SmarandacheNumber/Inline28.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="39" height="21" alt="n=1" />, 2, ..., the digit lengths <img src="/images/equations/SmarandacheNumber/Inline29.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="37" height="21" alt="D(n)" /> of <img src="/images/equations/SmarandacheNumber/Inline30.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="47" height="21" alt="Sm(n)" /> are therefore 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, ... (OEIS <a href="http://oeis.org/A058183">A058183</a>). </p> <div class="center-image"> <img src="/images/gifs/ConsecutiveIntegersBinary.jpg" class="" style="max-width:100%;max-height:100%;" width="510" alt="Plots of the concatenation of consecutive integers in base 2" /> </div> <p> The results of concatenating the binary representations of the first few integers are 1, 110, 11011, 11011100, 11011100101, ... (OEIS <a href="http://oeis.org/A058935">A058935</a>). These digit sequences are plotted above for <img src="/images/equations/SmarandacheNumber/Inline31.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="39" height="21" alt="n=1" /> to 90. Interpreting the digit sequence as a binary fraction, the result is the <a href="/BinaryChampernowneConstant.html">binary Champernowne constant</a> <img src="/images/equations/SmarandacheNumber/Inline32.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="19" height="21" alt="C_2" />. </p> <div class="center-image"> <img style="max-width:100%;max-height:100%;" width="545.722" src="images/eps-svg/ConsecutiveIntegersCumulativeSum_800.svg" class="" alt="ConsecutiveIntegersCumulativeSum" /> </div> <p> Interestingly, taking the <a href="/CumulativeSum.html">cumulative sum</a> <img src="/images/equations/SmarandacheNumber/Inline33.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="55" height="21" alt="2x_i-1" /> where <img src="/images/equations/SmarandacheNumber/Inline34.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="26" height="21" alt="{x_i}" /> are the digits <img src="/images/equations/SmarandacheNumber/Inline35.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="19" height="21" alt="C_2" /> gives a plot showing <a href="/Batrachion.html">batrachion</a>-like structure (left figure), and doing the same with <img src="/images/equations/SmarandacheNumber/Inline36.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="46" height="23" alt="{x_i}_(i=2)^infty" /> (right figure) gives structures resembling the <a href="/BlancmangeFunction.html">Blancmange function</a> (and the <a href="/Hofstadter-Conway10000-DollarSequence.html">Hofstadter-Conway $10,000 sequence</a>). </p> </div>  <hr class="margin-b-1-1-4"> <div class="c-777 entry-secondary-content">  <h2>See also</h2><a href="/ChampernowneConstant.html">Champernowne Constant</a>, <a href="/ChampernowneConstantDigits.html">Champernowne Constant Digits</a>, <a href="/ConsecutiveNumberSequences.html">Consecutive Number Sequences</a>, <a href="/IntegerSequencePrimes.html">Integer Sequence Primes</a>, <a href="/SmarandachePrime.html">Smarandache Prime</a>, <a href="/SmarandacheSequences.html">Smarandache Sequences</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="https://www.wolframalpha.com/input/?i=1200+-+450">1200 - 450</a></li> <li><a target="_blank" href="https://www.wolframalpha.com/input/?i=div+%28grad+f%29">div (grad f)</a></li> <li><a target="_blank" href="http://www.wolframalpha.com/input/?i=L+integral+transform">L integral transform</a></li> </ul> </div> </div>   <h2>References</h2><cite>--. "The Great Smarandache PRPrime search." <a href="http://smarandache.ddns.net:1200/server_stats.html">http://smarandache.ddns.net:1200/server_stats.html</a>.</cite><cite>Mudge, M. "Top of the Class." <i>Personal Computer World,</i> 674-675, June 1995.</cite><cite>Mudge, M. "Not Numerology but Numeralogy!" <i>Personal Computer World,</i> 279-280, 1997.</cite><cite>Sloane, N. J. A. Sequences <a href="http://oeis.org/A007908">A007908</a>, <a href="http://oeis.org/A058183">A058183</a>, and <a href="http://oeis.org/A058935">A058935</a> in "The On-Line Encyclopedia of Integer Sequences."</cite><cite>Smarandache, F. <i><a href="http://www.amazon.com/exec/obidos/ASIN/1879585006/ref=nosim/ericstreasuretro">Only Problems, Not Solutions!, 4th ed.</a></i> Phoenix, AZ: Xiquan, 1993.</cite><cite>Stephan, R. W. "Factors and Primes in Two Smarandache Sequences." <i>Smarandache Notions J.</i> <b>9</b>, 4-10, 1998.</cite><cite>Wolfram, S. <i><a href="http://www.amazon.com/exec/obidos/ASIN/1579550088/ref=nosim/ericstreasuretro">A New Kind of Science.</a></i> Champaign, IL: Wolfram Media, p. <a href="http://www.wolframscience.com/nksonline/page-913c-text">913</a>, 2002.</cite>   <h2>Cite this as:</h2> <p> <a href="/about/author.html">Weisstein, Eric W.</a> "Smarandache Number." 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