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A378222 - 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>A378222 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A378222" 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=%2fA378222">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="A378222 - 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> A378222 </div> <div class=seqname> Number of ordered factorizations of the odd part of n into factors > 1. </div> </div> <div class=scorerefs> 0 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 3, 1, 1, 1, 2, 1, 4, 1, 1, 3, 1, 1, 3, 1, 3, 2, 1, 1, 3, 1, 1, 3, 1, 1, 8, 1, 1, 1, 2, 2, 3, 1, 1, 4, 3, 1, 3, 1, 1, 3, 1, 1, 8, 1, 3, 3, 1, 1, 3, 3, 1, 2, 1, 1, 8, 1, 3, 3, 1, 1, 8, 1, 1, 3, 3, 1, 3, 1, 1, 8, 3, 1, 3, 1, 3, 1, 1, 2, 8, 2, 1, 3, 1, 1, 13</div> <div class=seqdatalinks> (<a href="/A378222/list">list</a>; <a href="/A378222/graph">graph</a>; <a href="/search?q=A378222+-id:A378222">refs</a>; <a href="/A378222/listen">listen</a>; <a href="/history?seq=A378222">history</a>; <a href="/search?q=id:A378222&fmt=text">text</a>; <a href="/A378222/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>1,9</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>Antti Karttunen, <a href="/A378222/b378222.txt">Table of n, a(n) for n = 1..20000</a></div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>a(n) = <a href="/A074206" title="Kalm谩r's [Kalmar's] problem: number of ordered factorizations of n.">A074206</a>(<a href="/A000265" title="Remove all factors of 2 from n; or largest odd divisor of n; or odd part of n.">A000265</a>(n)).</div> <div class=sectline>For all n >= 1, a(2n) = a(n).</div> <div class=sectline>For all n >= 0, a(2n+1) = <a href="/A352063" title="Number of ordered factorizations of 2*n + 1 into odd factors > 1.">A352063</a>(n) = <a href="/A002033" title="Number of perfect partitions of n.">A002033</a>(2*n) = <a href="/A074206" title="Kalm谩r's [Kalmar's] problem: number of ordered factorizations of n.">A074206</a>(2*n+1).</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(PARI)</div> <div class=sectline><a href="/A000265" title="Remove all factors of 2 from n; or largest odd divisor of n; or odd part of n.">A000265</a>(n) = (n>>valuation(n, 2));</div> <div class=sectline><a href="/A074206" title="Kalm谩r's [Kalmar's] problem: number of ordered factorizations of n.">A074206</a>(n) = if(n>1, sumdiv(n, i, if(i<n, <a href="/A074206" title="Kalm谩r's [Kalmar's] problem: number of ordered factorizations of n.">A074206</a>(i))), n); \\ From <a href="/A074206" title="Kalm谩r's [Kalmar's] problem: number of ordered factorizations of n.">A074206</a></div> <div class=sectline><a href="/A378222" title="Number of ordered factorizations of the odd part of n into factors > 1.">A378222</a>(n) = <a href="/A074206" title="Kalm谩r's [Kalmar's] problem: number of ordered factorizations of n.">A074206</a>(<a href="/A000265" title="Remove all factors of 2 from n; or largest odd divisor of n; or odd part of n.">A000265</a>(n));</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A000265" title="Remove all factors of 2 from n; or largest odd divisor of n; or odd part of n.">A000265</a>, <a href="/A074206" title="Kalm谩r's [Kalmar's] problem: number of ordered factorizations of n.">A074206</a>.</div> <div class=sectline>Bisections: <a href="/A352063" title="Number of ordered factorizations of 2*n + 1 into odd factors > 1.">A352063</a>, and the sequence itself.</div> <div class=sectline>Sequence in context: <a href="/A259396" title="Length of runs of identical terms in A080378.">A259396</a> <a href="/A328672" title="Number of integer partitions of n with relatively prime parts in which no two distinct parts are relatively prime.">A328672</a> <a href="/A368465" title="Number of even terms in each row of the iterates of the Christmas tree pattern map (A367508).">A368465</a> * <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="/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>Adjacent sequences: <a href="/A378219" title="Dirichlet inverse of A253249.">A378219</a> <a href="/A378220" title="Dirichlet inverse of phi(A003961(n)), where A003961 is fully multiplicative function with a(prime(i)) = prime(i+1).">A378220</a> <a href="/A378221" title="Dirichlet inverse of sigma(A003961(n)), where A003961 is fully multiplicative function with a(prime(i)) = prime(i+1).">A378221</a> * <a href="/A378228" title="Dirichlet inverse of A341528, where A341528(n) = n * sigma(A003961(n)), and A003961 is fully multiplicative with a(prime(i))...">A378228</a> <a href="/A378229" title="Dirichlet inverse of A341529, where A341529(n) = sigma(n) * A003961(n), and A003961 is fully multiplicative with a(prime(i))...">A378229</a> <a href="/A378231" title="Deficiency of prime-shifted squares: a(n) = 2*A003961(n^2) - sigma(A003961(n^2)), where A003961 is fully multiplicative func...">A378231</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="added within the last two weeks">new</span></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:Antti_Karttunen">Antti Karttunen</a>, Nov 24 2024</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 November 24 16:42 EST 2024. Contains 378083 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>