CINXE.COM
A328673 - 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>A328673 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A328673" 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=%2fA328673">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="A328673 - 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> A328673 </div> <div class=seqname> Number of integer partitions of n in which no two distinct parts are relatively prime. </div> </div> <div class=scorerefs> 31 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>1, 1, 2, 2, 3, 2, 5, 2, 6, 4, 9, 2, 15, 2, 17, 10, 23, 2, 39, 2, 46, 18, 58, 2, 95, 8, 103, 31, 139, 2, 219, 3, 232, 59, 299, 22, 452, 4, 492, 104, 645, 5, 920, 5, 1006, 204, 1258, 8, 1785, 21, 1994, 302, 2442, 11, 3366, 71, 3738, 497, 4570, 18, 6253, 24, 6849</div> <div class=seqdatalinks> (<a href="/A328673/list">list</a>; <a href="/A328673/graph">graph</a>; <a href="/search?q=A328673+-id:A328673">refs</a>; <a href="/A328673/listen">listen</a>; <a href="/history?seq=A328673">history</a>; <a href="/search?q=id:A328673&fmt=text">text</a>; <a href="/A328673/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>0,3</div> </div> </div> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <div class=sectline>A partition with no two distinct parts relatively prime is said to be intersecting.</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>Fausto A. C. Cariboni, <a href="/A328673/b328673.txt">Table of n, a(n) for n = 0..350</a></div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>a(n > 0) = <a href="/A200976" title="Number of partitions of n such that each pair of parts (if any) has a common factor.">A200976</a>(n) + 1.</div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>The a(1) = 1 through a(10) = 9 partitions (A = 10):</div> <div class=sectline> 1 2 3 4 5 6 7 8 9 A</div> <div class=sectline> 11 111 22 11111 33 1111111 44 63 55</div> <div class=sectline> 1111 42 62 333 64</div> <div class=sectline> 222 422 111111111 82</div> <div class=sectline> 111111 2222 442</div> <div class=sectline> 11111111 622</div> <div class=sectline> 4222</div> <div class=sectline> 22222</div> <div class=sectline> 1111111111</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>Table[Length[Select[IntegerPartitions[n], And@@(GCD[##]>1&)@@@Subsets[Union[#], {2}]&]], {n, 0, 20}]</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>The Heinz numbers of these partitions are <a href="/A328867" title="Heinz numbers of integer partitions in which no two distinct parts are relatively prime.">A328867</a> (strict case is <a href="/A318719" title="Heinz numbers of strict integer partitions in which no two parts are relatively prime.">A318719</a>).</div> <div class=sectline>The relatively prime case is <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>.</div> <div class=sectline>The strict case is <a href="/A318717" title="Number of strict integer partitions of n in which no two parts are relatively prime.">A318717</a>.</div> <div class=sectline>The version for non-isomorphic multiset partitions is <a href="/A319752" title="Number of non-isomorphic intersecting multiset partitions of weight n.">A319752</a>.</div> <div class=sectline>The version for set-systems is <a href="/A305843" title="Number of labeled spanning intersecting set-systems on n vertices.">A305843</a>.</div> <div class=sectline>The version involving all parts (not just distinct ones) is <a href="/A200976" title="Number of partitions of n such that each pair of parts (if any) has a common factor.">A200976</a>.</div> <div class=sectline>Cf. <a href="/A000837" title="Number of partitions of n into relatively prime parts. Also aperiodic partitions.">A000837</a>, <a href="/A202425" title="Number of partitions of n into parts having pairwise common factors but no overall common factor.">A202425</a>, <a href="/A305148" title="Number of integer partitions of n whose distinct parts are pairwise indivisible.">A305148</a>, <a href="/A305854" title="Number of unlabeled spanning intersecting set-systems on n vertices.">A305854</a>, <a href="/A306006" title="Number of non-isomorphic intersecting set-systems of weight n.">A306006</a>, <a href="/A316476" title="Stable numbers. Numbers whose distinct prime indices are pairwise indivisible.">A316476</a>, <a href="/A326910" title="BII-numbers of pairwise intersecting set-systems.">A326910</a>.</div> <div class=sectline>Sequence in context: <a href="/A363724" title="Number of integer partitions of n whose mean is a mode, i.e., partitions whose mean appears at least as many times as each o...">A363724</a> <a href="/A345268" title="a(n) = Sum_{d|n} d^(phi(n/d) - 1).">A345268</a> <a href="/A164941" title="a(n) = Sum_{d|n} phi(n/d)^(d-1).">A164941</a> * <a href="/A115119" title="Number of imprimitive (periodic) n-bead necklaces with beads of 2 colors when turning over is allowed.">A115119</a> <a href="/A066656" title="a(n) = A000031(n) - A001037(n).">A066656</a> <a href="/A164896" title="Number of subsets (up to cyclic shifts) of the n-th roots of 1 with zero sum.">A164896</a></div> <div class=sectline>Adjacent sequences: <a href="/A328670" title="Number of aperiodic compositions of n where every pair of adjacent parts (including the last with the first) is relatively p...">A328670</a> <a href="/A328671" title="Numbers whose binary indices are relatively prime and pairwise indivisible.">A328671</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="/A328674" title="Numbers whose distinct prime indices have no consecutive divisible parts.">A328674</a> <a href="/A328675" title="Number of integer partitions of n with no two distinct consecutive parts divisible.">A328675</a> <a href="/A328676" title="Number of relatively prime integer partitions of n whose distinct parts are pairwise indivisible.">A328676</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:Gus_Wiseman">Gus Wiseman</a>, Oct 29 2019</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 March 9 11:53 EDT 2025. Contains 381557 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>