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A328672 - 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>A328672 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A328672" 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=%2fA328672">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="A328672 - 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> A328672 </div> <div class=seqname> Number of integer partitions of n with relatively prime parts in which no two distinct parts are relatively prime. </div> </div> <div class=scorerefs> 7 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 4, 1, 4, 1, 1, 2, 7, 1, 6, 1, 3, 3, 10, 1, 9, 3, 5, 4, 17, 1, 23, 6, 7, 6, 20, 3, 36, 9, 15, 7, 45, 5, 56, 14, 17, 20, 65, 7, 83, 18, 40</div> <div class=seqdatalinks> (<a href="/A328672/list">list</a>; <a href="/A328672/graph">graph</a>; <a href="/search?q=A328672+-id:A328672">refs</a>; <a href="/A328672/listen">listen</a>; <a href="/history?seq=A328672">history</a>; <a href="/search?q=id:A328672&fmt=text">text</a>; <a href="/A328672/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>0,32</div> </div> </div> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <div class=sectline>Positions of terms greater than 1 are {31, 37, 41, 43, 46, 47, 49, ...}.</div> <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="/A328672/b328672.txt">Table of n, a(n) for n = 0..400</a></div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>a(n > 0) = <a href="/A202425" title="Number of partitions of n into parts having pairwise common factors but no overall common factor.">A202425</a>(n) + 1.</div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>Examples:</div> <div class=sectline> a(31) = 2: a(46) = 2:</div> <div class=sectline> (15,10,6) (15,15,10,6)</div> <div class=sectline> (1^31) (1^46)</div> <div class=sectline> a(37) = 3: a(47) = 7:</div> <div class=sectline> (15,12,10) (20,15,12)</div> <div class=sectline> (15,10,6,6) (21,14,12)</div> <div class=sectline> (1^37) (20,15,6,6)</div> <div class=sectline> a(41) = 4: (21,14,6,6)</div> <div class=sectline> (20,15,6) (15,12,10,10)</div> <div class=sectline> (21,14,6) (15,10,10,6,6)</div> <div class=sectline> (15,10,10,6) (1^47)</div> <div class=sectline> (1^41) a(49) = 6:</div> <div class=sectline> a(43) = 4: (24,15,10)</div> <div class=sectline> (18,15,10) (18,15,10,6)</div> <div class=sectline> (15,12,10,6) (15,12,12,10)</div> <div class=sectline> (15,10,6,6,6) (15,12,10,6,6)</div> <div class=sectline> (1^43) (15,10,6,6,6,6)</div> <div class=sectline> (1^39)</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>Table[Length[Select[IntegerPartitions[n], GCD@@#==1&&And[And@@(GCD[##]>1&)@@@Subsets[Union[#], {2}]]&]], {n, 0, 32}]</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="/A328679" title="Heinz numbers of integer partitions with no two distinct parts relatively prime, but with no divisor in common to all of the...">A328679</a>.</div> <div class=sectline>The strict case is <a href="/A318715" title="Number of strict integer partitions of n with relatively prime parts in which no two parts are relatively prime.">A318715</a>.</div> <div class=sectline>The version for non-isomorphic multiset partitions is <a href="/A319759" title="Number of non-isomorphic intersecting multiset partitions of weight n with empty intersection.">A319759</a>.</div> <div class=sectline>Relatively prime partitions are <a href="/A000837" title="Number of partitions of n into relatively prime parts. Also aperiodic partitions.">A000837</a>.</div> <div class=sectline>Intersecting partitions are <a href="/A328673" title="Number of integer partitions of n in which no two distinct parts are relatively prime.">A328673</a>.</div> <div class=sectline>Cf. <a href="/A078374" title="Number of partitions of n into distinct and relatively prime parts.">A078374</a>, <a href="/A285573" title="Number of finite nonempty sets of pairwise indivisible divisors of n.">A285573</a>, <a href="/A289509" title="Numbers k such that the gcd of the indices j for which the j-th prime prime(j) divides k is 1.">A289509</a>, <a href="/A291166" title="Connected Haar graph numbers.">A291166</a>, <a href="/A303140" title="Number of strict integer partitions of n with at least two but not all parts having a common divisor greater than 1.">A303140</a>, <a href="/A305148" title="Number of integer partitions of n whose distinct parts are pairwise indivisible.">A305148</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>, <a href="/A326912" title="BII-numbers of pairwise intersecting set-systems with empty intersection.">A326912</a>.</div> <div class=sectline>Sequence in context: <a href="/A101446" title="a(n)= least k such that n^k*(n^k-1)-1 is prime with n > 1.">A101446</a> <a href="/A333769" title="Irregular triangle read by rows where row k is the sequence of run-lengths of the k-th composition in standard order.">A333769</a> <a href="/A259396" title="Length of runs of identical terms in A080378.">A259396</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="/A378222" title="Number of ordered factorizations of the odd part of n into factors > 1.">A378222</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></div> <div class=sectline>Adjacent sequences: <a href="/A328669" title="Number of Lyndon compositions of n where every pair of adjacent parts (including the last with the first) is relatively prime.">A328669</a> <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="/A328673" title="Number of integer partitions of n in which no two distinct parts are relatively prime.">A328673</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></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 8 21:53 EST 2025. Contains 381553 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>