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A062760 - 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>A062760 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A062760" 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=%2fA062760">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="A062760 - 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> A062760 </div> <div class=seqname> a(n) is n divided by the largest power of the squarefree kernel of n (<a href="/A007947" title="Largest squarefree number dividing n: the squarefree kernel of n, rad(n), radical of n.">A007947</a>) which divides it. </div> </div> <div class=scorerefs> 7 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 2, 3, 1, 1, 8, 1, 5, 1, 2, 1, 9, 1, 4, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 5, 2, 1, 1, 1, 8, 1, 1, 1, 2, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 16, 1, 7, 3, 1, 1, 1, 1, 4</div> <div class=seqdatalinks> (<a href="/A062760/list">list</a>; <a href="/A062760/graph">graph</a>; <a href="/search?q=A062760+-id:A062760">refs</a>; <a href="/A062760/listen">listen</a>; <a href="/history?seq=A062760">history</a>; <a href="/search?q=id:A062760&fmt=text">text</a>; <a href="/A062760/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) divides <a href="/A003557" title="n divided by largest squarefree divisor of n; if n = Product p(k)^e(k) then a(n) = Product p(k)^(e(k)-1), with a(1) = 1.">A003557</a> but is not equal to it.</div> <div class=sectline>a(n) is least d such that the prime power exponents of n/d are all equal; see also <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>. - <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="/A062760/b062760.txt">Table of n, a(n) for n = 1..10000</a></div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>a(n) = 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="/A051904" title="Minimal exponent in prime factorization of n.">A051904</a>(n)).</div> <div class=sectline>a(n) = n/<a href="/A062759" title="Largest power of squarefree kernel of n (= A007947) which divides n.">A062759</a>(n). - <a href="/wiki/User:Amiram_Eldar">Amiram Eldar</a>, Feb 12 2023</div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>n=1800: the squarefree kernel is 2*3*5 = 30 and 900 = 30^2 divides n, a(1800) = 2, the quotient of 1800/900.</div> </div> </div> <div class=section> <div class=sectname>MAPLE</div> <div class=sectbody> <div class=sectline>f:= proc(n) local F, m, t;</div> <div class=sectline> F:= ifactors(n)[2];</div> <div class=sectline> m:= min(seq(t[2], t=F));</div> <div class=sectline> mul(t[1]^(t[2]-m), t=F)</div> <div class=sectline>end proc:</div> <div class=sectline>map(f, [$1..200]); # <a href="/wiki/User:Robert_Israel">Robert Israel</a>, Nov 03 2017</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>{1}~Join~Table[n/#^IntegerExponent[n, #] &@ Last@ Select[Divisors@ n, SquareFreeQ], {n, 2, 104}] (* <a href="/wiki/User:Michael_De_Vlieger">Michael De Vlieger</a>, Nov 02 2017 *)</div> <div class=sectline>a[n_] := Module[{f = FactorInteger[n], e}, e = Min[f[[;; , 2]]]; f[[;; , 2]] -= e; Times @@ Power @@@ f]; Array[a, 100] (* <a href="/wiki/User:Amiram_Eldar">Amiram Eldar</a>, Feb 12 2023 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(PARI)</div> <div class=sectline><a href="/A007947" title="Largest squarefree number dividing n: the squarefree kernel of n, rad(n), radical of n.">A007947</a>(n) = factorback(factorint(n)[, 1]); \\ <a href="/wiki/User:Andrew_Lelechenko">Andrew Lelechenko</a>, May 09 2014</div> <div class=sectline><a href="/A051904" title="Minimal exponent in prime factorization of n.">A051904</a>(n) = if(1==n, 0, vecmin(factor(n)[, 2])); \\ After <a href="/wiki/User:Charles_R_Greathouse_IV">Charles R Greathouse IV</a>'s code</div> <div class=sectline><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>(n) = 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="/A051904" title="Minimal exponent in prime factorization of n.">A051904</a>(n)); \\ <a href="/wiki/User:Antti_Karttunen">Antti Karttunen</a>, Sep 23 2017</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A001694" title="Powerful numbers, definition (1): if a prime p divides n then p^2 must also divide n (also called squareful, square full, sq...">A001694</a>, <a href="/A003557" title="n divided by largest squarefree divisor of n; if n = Product p(k)^e(k) then a(n) = Product p(k)^(e(k)-1), with a(1) = 1.">A003557</a>, <a href="/A007947" title="Largest squarefree number dividing n: the squarefree kernel of n, rad(n), radical of n.">A007947</a>, <a href="/A051904" title="Minimal exponent in prime factorization of n.">A051904</a>, <a href="/A003557" title="n divided by largest squarefree divisor of n; if n = Product p(k)^e(k) then a(n) = Product p(k)^(e(k)-1), with a(1) = 1.">A003557</a>, <a href="/A052485" title="Weak numbers (i.e., not powerful (1)): there is a prime p where p|n is true but p^2|n is not true.">A052485</a>, <a href="/A062759" title="Largest power of squarefree kernel of n (= A007947) which divides n.">A062759</a>.</div> <div class=sectline>Cf. <a href="/A059404" title="Numbers with different exponents in their prime factorizations.">A059404</a> (n such that a(n)>1), <a href="/A072774" title="Powers of squarefree numbers.">A072774</a> (n such that a(n)=1).</div> <div class=sectline>Cf. <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>.</div> <div class=sectline>Sequence in context: <a href="/A325355" title="One plus the number of steps applying A325351 (Heinz number of augmented differences of reversed prime indices) to reach a f...">A325355</a> <a href="/A219093" title="Denominator of the least reduced fraction b/c > 1 using divisors b and c of n.">A219093</a> <a href="/A380442" title="Second coefficient of the lindep transform of phi(n).">A380442</a> * <a href="/A323163" title="Greatest common divisor of product (1+(p^e)) and product p^(e-1), where p ranges over prime factors of n, with e correspondi...">A323163</a> <a href="/A322318" title="a(n) = gcd(A003557(n), A048250(n)).">A322318</a> <a href="/A014649" title="Number of partitions of n into its nonprime power divisors with at least one part of size 1.">A014649</a></div> <div class=sectline>Adjacent sequences: <a href="/A062757" title="Denominator of sum of first n terms of the series 1/15 + 1/63 + 1/80 ... in which the denominators are perfect squares - 1 w...">A062757</a> <a href="/A062758" title="Product of squares of divisors of n.">A062758</a> <a href="/A062759" title="Largest power of squarefree kernel of n (= A007947) which divides n.">A062759</a> * <a href="/A062761" title="Number of powerful numbers between 2^(n-1)+1 and 2^n.">A062761</a> <a href="/A062762" title="Number of powerful numbers not exceeding 2^n.">A062762</a> <a href="/A062763" title="a(n) is the greatest common divisor of (n-1)! and n^n.">A062763</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></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:Labos_Elemer">Labos Elemer</a>, Jul 16 2001</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 16 09:55 EST 2025. Contains 380897 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>