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A066636 - 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>A066636 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A066636" 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=%2fA066636">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="A066636 - 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> A066636 </div> <div class=seqname> a(n) = <a href="/A066638" title="Smallest power of a squarefree number that is a multiple of n.">A066638</a>(n)/n, where <a href="/A066638" title="Smallest power of a squarefree number that is a multiple of n.">A066638</a>(n) is the smallest power of a squarefree kernel of n that is a multiple of n. </div> </div> <div class=scorerefs> 6 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 5, 1, 1, 1, 9, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 25, 1, 1, 1, 11, 5, 1, 1, 27, 1, 2, 1, 13, 1, 4, 1, 49, 1, 1, 1, 15, 1, 1, 7, 1, 1, 1, 1, 17, 1, 1, 1, 3, 1, 1, 3, 19, 1, 1, 1, 125, 1, 1, 1, 21, 1</div> <div class=seqdatalinks> (<a href="/A066636/list">list</a>; <a href="/A066636/graph">graph</a>; <a href="/search?q=A066636+-id:A066636">refs</a>; <a href="/A066636/listen">listen</a>; <a href="/history?seq=A066636">history</a>; <a href="/search?q=id:A066636&fmt=text">text</a>; <a href="/A066636/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>1,12</div> </div> </div> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <div class=sectline>a(n) is the least m such that the prime power exponents of m*n are all equal; see also <a href="/A062760" title="a(n) is n divided by the largest power of the squarefree kernel of n (A007947) which divides it.">A062760</a>. - <a href="/wiki/User:David_James_Sycamore">David James Sycamore</a>, Jun 13 2024</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>Antti Karttunen, <a href="/A066636/b066636.txt">Table of n, a(n) for n = 1..16384</a></div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>a(n) = (<a href="/A007947" title="Largest squarefree number dividing n: the squarefree kernel of n, rad(n), radical of n.">A007947</a>(n)^<a href="/A051903" title="Maximum exponent in the prime factorization of n.">A051903</a>(n))/n. - <a href="/wiki/User:Antti_Karttunen">Antti Karttunen</a>, Nov 20 2017</div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>12 = 2^2*3^1 so m = 3 (3*12 = 36 = 2^2*3^2).</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>Array[Apply[Times, #2[[All, 1]]]^Max[#2[[All, -1]] ]/#1 & @@ {#, FactorInteger@ #} &, 85] (* <a href="/wiki/User:Michael_De_Vlieger">Michael De Vlieger</a>, Nov 20 2017 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(PARI)</div> <div class=sectline><a href="/A066638" title="Smallest power of a squarefree number that is a multiple of n.">A066638</a>(n) = { if(n==1, return(1)); my(f=factor(n), me=vecmax(f[, 2])); (prod(i=1, #f~, f[i, 1])^me); }; \\ After <a href="/wiki/User:Charles_R_Greathouse_IV">Charles R Greathouse IV</a>'s code.</div> <div class=sectline><a href="/A066636" title="a(n) = A066638(n)/n, where A066638(n) is the smallest power of a squarefree kernel of n that is a multiple of n.">A066636</a>(n) = (<a href="/A066638" title="Smallest power of a squarefree number that is a multiple of n.">A066638</a>(n)/n); \\ <a href="/wiki/User:Antti_Karttunen">Antti Karttunen</a>, Nov 20 2017</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A007947" title="Largest squarefree number dividing n: the squarefree kernel of n, rad(n), radical of n.">A007947</a>, <a href="/A051903" title="Maximum exponent in the prime factorization of n.">A051903</a>, <a href="/A066638" title="Smallest power of a squarefree number that is a multiple of n.">A066638</a>.</div> <div class=sectline>Cf. <a href="/A062760" title="a(n) is n divided by the largest power of the squarefree kernel of n (A007947) which divides it.">A062760</a>.</div> <div class=sectline>Sequence in context: <a href="/A348969" title="a(n) = n / gcd(n, A099377(n)).">A348969</a> <a href="/A030380" title="Length of n-th run of digit 1 in A030373.">A030380</a> <a href="/A375932" title="The largest unitary k-free divisor of n where k = A051903(n) is the maximum exponent in the prime factorization of n.">A375932</a> * <a href="/A137578" title="Smallest of the least frequent digits in decimal representation of n!.">A137578</a> <a href="/A200146" title="Triangle read by rows: T(n, k) = mod(k^(n - 1), n), where 1 <= k < n.">A200146</a> <a href="/A319095" title="Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1, k) + T(n-1, k-1) + T(n-1, k-2) + 3*T(n-1, k-3) + T(n-1, k-4) + T(n-1, k-5...">A319095</a></div> <div class=sectline>Adjacent sequences: <a href="/A066633" title="Triangle T(n,k), n >= 1, 1 <= k <= n, giving number of k's in all partitions of n.">A066633</a> <a href="/A066634" title="Number of triangulations of the cyclic polytope C(n, n-5).">A066634</a> <a href="/A066635" title="Distance from n to closest square different from n.">A066635</a> * <a href="/A066637" title="Total number of elements in all factorizations of n with all factors > 1.">A066637</a> <a href="/A066638" title="Smallest power of a squarefree number that is a multiple of n.">A066638</a> <a href="/A066639" title="Number of partitions of n with floor(n/2) parts.">A066639</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="sequence has an interesting graph"><a href="/A066636/graph">look</a></span></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Jan 09 2002</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 February 21 07:20 EST 2025. Contains 381068 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>