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<!doctype html> <html lang="en" class="foundationsofmathematics mathworldcontributors numbertheory"> <head> <title>Powerful Number -- from Wolfram MathWorld</title> <meta name="DC.Title" content="Powerful Number" /> <meta name="DC.Creator" content="Weisstein, Eric W." /> <meta name="DC.Description" content="An integer m such that if p|m, then p^2|m, is called a powerful number. There are an infinite number of powerful numbers, and the first few are 1, 4, 8, 9, 16, 25, 27, 32, 36, 49, ... (OEIS A001694). Powerful numbers are always of the form a^2b^3 for a,b>=1. The numbers of powerful numbers <=10, 10^2, 10^3, ... are given by 4, 14, 54, 185, 619, 2027, 6553, 21044, 67231, 214122, 680330, 2158391, ... (OEIS A118896). Golomb (1970) showed that the sum over the reciprocals of the powerful..." /> <meta name="description" content="An integer m such that if p|m, then p^2|m, is called a powerful number. There are an infinite number of powerful numbers, and the first few are 1, 4, 8, 9, 16, 25, 27, 32, 36, 49, ... (OEIS A001694). Powerful numbers are always of the form a^2b^3 for a,b>=1. The numbers of powerful numbers <=10, 10^2, 10^3, ... are given by 4, 14, 54, 185, 619, 2027, 6553, 21044, 67231, 214122, 680330, 2158391, ... (OEIS A118896). Golomb (1970) showed that the sum over the reciprocals of the powerful..." /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2005-11-30" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2006-05-04" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:Number Theory:Special Numbers:Divisor-Related Numbers" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:Foundations of Mathematics:Mathematical Problems:Unsolved Problems" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:MathWorld Contributors:Noe" /> <meta name="DC.Rights" content="Copyright 1999-2024 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/PowerfulNumber.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="2006-05-04" /> <meta property="og:image" content="https://mathworld.wolfram.com/images/socialmedia/share/ogimage_PowerfulNumber.png"> <meta property="og:url" content="https://mathworld.wolfram.com/PowerfulNumber.html"> <meta property="og:type" content="website"> <meta property="og:title" content="Powerful Number -- from Wolfram MathWorld"> <meta property="og:description" content="An integer m such that if p|m, then p^2|m, is called a powerful number. There are an infinite number of powerful numbers, and the first few are 1, 4, 8, 9, 16, 25, 27, 32, 36, 49, ... (OEIS A001694). Powerful numbers are always of the form a^2b^3 for a,b>=1. The numbers of powerful numbers <=10, 10^2, 10^3, ... are given by 4, 14, 54, 185, 619, 2027, 6553, 21044, 67231, 214122, 680330, 2158391, ... (OEIS A118896). Golomb (1970) showed that the sum over the reciprocals of the powerful..."> <meta name="twitter:card" content="summary_large_image"> <meta name="twitter:site" content="@WolframResearch"> <meta name="twitter:title" content="Powerful Number -- from Wolfram MathWorld"> <meta name="twitter:description" content="An integer m such that if p|m, then p^2|m, is called a powerful number. There are an infinite number of powerful numbers, and the first few are 1, 4, 8, 9, 16, 25, 27, 32, 36, 49, ... (OEIS A001694). Powerful numbers are always of the form a^2b^3 for a,b>=1. The numbers of powerful numbers <=10, 10^2, 10^3, ... are given by 4, 14, 54, 185, 619, 2027, 6553, 21044, 67231, 214122, 680330, 2158391, ... (OEIS A118896). Golomb (1970) showed that the sum over the reciprocals of the powerful..."> <meta name="twitter:image:src" content="https://mathworld.wolfram.com/images/socialmedia/share/ogimage_PowerfulNumber.png"> <link rel="canonical" href="https://mathworld.wolfram.com/PowerfulNumber.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/Divisor-RelatedNumbers.html">Divisor-Related Numbers</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/FoundationsofMathematics.html">Foundations of Mathematics</a> </li> <li> <a href="/topics/MathematicalProblems.html">Mathematical Problems</a> </li> <li> <a href="/topics/UnsolvedProblems.html">Unsolved Problems</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/MathWorldContributors.html">MathWorld Contributors</a> </li> <li> <a href="/topics/Noe.html">Noe</a> </li> </ul></nav>   <h1>Powerful Number</h1>  <hr class="margin-t-1-8 margin-b-3-4">  <div class="attachments text-align-r"> <a href="/notebooks/NumberTheory/PowerfulNumber.nb" download="PowerfulNumber.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> An <a href="/Integer.html">integer</a> <img src="/images/equations/PowerfulNumber/Inline1.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="m" /> such that if <img src="/images/equations/PowerfulNumber/Inline2.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="36" height="24" alt="p|m" />, then <img src="/images/equations/PowerfulNumber/Inline3.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="43" height="25" alt="p^2|m" />, is called a powerful number. There are an infinite number of powerful numbers, and the first few are 1, 4, 8, 9, 16, 25, 27, 32, 36, 49, ... (OEIS <a href="http://oeis.org/A001694">A001694</a>). Powerful numbers are always <a href="/OftheForm.html">of the form</a> <img src="/images/equations/PowerfulNumber/Inline4.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="36" height="21" alt="a^2b^3" /> for <img src="/images/equations/PowerfulNumber/Inline5.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="56" height="21" alt="a,b>=1" />. </p> <p> The numbers of powerful numbers <img src="/images/equations/PowerfulNumber/Inline6.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="37" height="21" alt="<=10" />, <img src="/images/equations/PowerfulNumber/Inline7.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="25" height="21" alt="10^2" />, <img src="/images/equations/PowerfulNumber/Inline8.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="25" height="21" alt="10^3" />, ... are given by 4, 14, 54, 185, 619, 2027, 6553, 21044, 67231, 214122, 680330, 2158391, ... (OEIS <a href="http://oeis.org/A118896">A118896</a>). </p> <p> Golomb (1970) showed that the sum over the reciprocals of the powerful numbers <img src="/images/equations/PowerfulNumber/Inline9.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="31" height="21" alt="{P_k}" /> is 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/PowerfulNumber/Inline10.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="46" height="48" alt="sum_(k)1/(P_k)" /></td><td align="center" width="14"><img src="/images/equations/PowerfulNumber/Inline11.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/PowerfulNumber/Inline12.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="69" height="42" alt="(zeta(2)zeta(3))/(zeta(6))" /></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/PowerfulNumber/Inline13.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/PowerfulNumber/Inline14.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/PowerfulNumber/Inline15.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="91" height="20" alt="1.9435964..." /></td><td align="right" width="10"> <div id="eqn2" class="eqnum"> (2) </div> </td></tr> </table> </div> <p> (OEIS <a href="http://oeis.org/A082695">A082695</a>), where <img src="/images/equations/PowerfulNumber/Inline16.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="30" height="21" alt="zeta(z)" /> is the <a href="/RiemannZetaFunction.html">Riemann zeta function</a>. </p> <p> Not every <a href="/NaturalNumber.html">natural number</a> is the sum of two powerful numbers, but Heath-Brown (1988) has shown that every sufficiently large <a href="/NaturalNumber.html">natural number</a> is the sum of at most three powerful numbers. There are infinitely many pairs of consecutive powerful numbers, the first few being (8, 9), (288, 289), (675, 676), (9800, 9801), ... (OEIS <a href="http://oeis.org/A060355">A060355</a> and <a href="http://oeis.org/A118893">A118893</a>). </p> <p> Erdős (1975/1965) conjectured that there do not exist three consecutive powerful numbers. Golomb (1970) also considered this question, as did Mollin and Walsh (1986). The <a href="/Conjecture.html">conjecture</a> that there are no powerful number triples implies that there are infinitely many non-<a href="/WieferichPrime.html">Wieferich primes</a> (Granville 1986; Ribenboim 1989, p. 341; Vardi 1991). </p> <p> A separate usage of the term powerful number is for numbers which are the sums of <i>any</i> positive powers of their digits (not necessarily the same for each digit). The first few are 1, 2, 3, 4, 5, 6, 7, 8, 9, 24, 43, 63, 89, ... (OEIS <a href="http://oeis.org/A007532">A007532</a>). These are also called handsome numbers by Rivera, and are a special case of the <a href="/NarcissisticNumber.html">narcissistic numbers</a>. Powerful numbers representable in two distinct ways (<i>not</i> counting different powers of duplicated digits as distinct) are 264, 373, 375, 2132, 2223, 2241, 2243, 2245, 2263, (OEIS <a href="http://oeis.org/A050240">A050240</a>). Powerful numbers representable in two distinct ways (counting different powers of duplicated digits as distinct) are 224, 226, 264, 332, 334, 375, 377, 445, (OEIS <a href="http://oeis.org/A050241">A050241</a>). </p> </div>  <hr class="margin-b-1-1-4"> <div class="c-777 entry-secondary-content">  <h2>See also</h2><a href="/AchillesNumber.html">Achilles Number</a>, <a href="/NarcissisticNumber.html">Narcissistic Number</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=%7B25%2C+35%2C+10%2C+17%2C+29%2C+14%2C+21%2C+31%7D">{25, 35, 10, 17, 29, 14, 21, 31}</a></li> <li><a target="_blank" href="https://www.wolframalpha.com/input/?i=express+4.8675+through+pi+and+e">express 4.8675 through pi and e</a></li> <li><a target="_blank" href="http://www.wolframalpha.com/input/?i=hough+transform+of+t-rex+image">hough transform of t-rex image</a></li> </ul> </div> </div>   <h2>References</h2><cite>Erdős, P. "Problems and Results on Consecutive Integers." <i>Eureka</i> <b>38</b>, 3-8, 1975/1976.</cite><cite>Erdős, P. "Problems and Results on Consecutive Integers." <i>Publ. Math. Debrecen</i> <b>23</b>, 271-282, 1976.</cite><cite>Golomb, S. W. "Powerful Numbers." <i>Amer. Math. Monthly</i> <b>77</b>, 848-855, 1970.</cite><cite>Granville, A. "Powerful Numbers and Fermat's Last Theorem." <i>C. R. Math. Rep. Acad. Sci. Canada</i> <b>8</b>, 215-218, 1986.</cite><cite>Guy, R. K. "Powerful Numbers." §B16 in <i><a href="http://www.amazon.com/exec/obidos/ASIN/0387208607/ref=nosim/ericstreasuretro">Unsolved Problems in Number Theory, 2nd ed.</a></i> New York: Springer-Verlag, pp. 67-73, 1994.</cite><cite>Heath-Brown, D. R. "Ternary Quadratic Forms and Sums of Three Square-Full Numbers." In <i><a href="http://www.amazon.com/exec/obidos/ASIN/0817634142/ref=nosim/ericstreasuretro">Séminaire de Theorie des Nombres, Paris 1986-87</a></i> (Ed. C. Goldstein). Boston, MA: Birkhäuser, pp. 137-163, 1988.</cite><cite>Mollin, R. A. "The Power of Powerful Numbers." <i>Int. J. Math. Math. Sci.</i> <b>10</b>, 125-130, 1986. <a href="http://www.math.ucalgary.ca/~ramollin/PPN.pdf">http://www.math.ucalgary.ca/~ramollin/PPN.pdf</a>.</cite><cite>Mollin, R. and Walsh, P. "On Powerful Numbers." <i>Int. J. Math. Math. Sci.</i> <b>9</b>, 801-806, 1986.</cite><cite>Ribenboim, P. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0387944575/ref=nosim/ericstreasuretro">The New Book of Prime Number Records.</a></i> New York: Springer-Verlag, 1989.</cite><cite>Ribenboim, P. "Catalan's Conjecture." <i>Amer. Math. Monthly</i> <b>103</b>, 529-538, 1996.</cite><cite>Rivera, C. "Problems & Puzzles: Puzzle 015-Narcissistic and Handsome Primes." <a href="http://www.primepuzzles.net/puzzles/puzz_015.htm">http://www.primepuzzles.net/puzzles/puzz_015.htm</a>.</cite><cite>Sloane, N. J. A. Sequences <a href="http://oeis.org/A001694">A001694</a>/M3325, <a href="http://oeis.org/A007532">A007532</a>/M0487, <a href="http://oeis.org/A050240">A050240</a>, <a href="http://oeis.org/A050241">A050241</a>, <a href="http://oeis.org/A060355">A060355</a>, <a href="http://oeis.org/A082695">A082695</a>, <a href="http://oeis.org/A118893">A118893</a>, and <a href="http://oeis.org/A118896">A118896</a> in "The On-Line Encyclopedia of Integer Sequences."</cite><cite>Vardi, I. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0685479412/ref=nosim/ericstreasuretro">Computational Recreations in Mathematica.</a></i> Reading, MA: Addison-Wesley, pp. 59-62, 1991.</cite><h2>Referenced on Wolfram|Alpha</h2><a href="http://www.wolframalpha.com/entities/mathworld/powerful_number/e0/l9/z0/" title="Powerful Number" target="_blank">Powerful Number</a>   <h2>Cite this as:</h2> <p> <a href="/about/author.html">Weisstein, Eric W.</a> "Powerful Number." 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