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A349059 - OEIS
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href="/login?redirect=%2fA349059">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="A349059 - 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> A349059 </div> <div class=seqname> Number of weakly alternating ordered factorizations of n. </div> </div> <div class=scorerefs> 17 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>1, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 8, 1, 3, 3, 8, 1, 8, 1, 8, 3, 3, 1, 18, 2, 3, 4, 8, 1, 11, 1, 16, 3, 3, 3, 22, 1, 3, 3, 18, 1, 11, 1, 8, 8, 3, 1, 38, 2, 8, 3, 8, 1, 18, 3, 18, 3, 3, 1, 32, 1, 3, 8, 28, 3, 11, 1, 8, 3, 11, 1, 56, 1, 3, 8, 8, 3, 11, 1, 38, 8, 3</div> <div class=seqdatalinks> (<a href="/A349059/list">list</a>; <a href="/A349059/graph">graph</a>; <a href="/search?q=A349059+-id:A349059">refs</a>; <a href="/A349059/listen">listen</a>; <a href="/history?seq=A349059">history</a>; <a href="/search?q=id:A349059&fmt=text">text</a>; <a href="/A349059/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>1,4</div> </div> </div> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <div class=sectline>An ordered factorization of n is a finite sequence of positive integers > 1 with product n.</div> <div class=sectline>We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either.</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline><a href="/A349059/b349059.txt">Table of n, a(n) for n=1..82.</a></div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>a(2^n) = <a href="/A349052" title="Number of weakly alternating compositions of n.">A349052</a>(n).</div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>The ordered factorizations for n = 2, 4, 6, 8, 12, 24, 30:</div> <div class=sectline> (2) (4) (6) (8) (12) (24) (30)</div> <div class=sectline> (2*2) (2*3) (2*4) (2*6) (3*8) (5*6)</div> <div class=sectline> (3*2) (4*2) (3*4) (4*6) (6*5)</div> <div class=sectline> (2*2*2) (4*3) (6*4) (10*3)</div> <div class=sectline> (6*2) (8*3) (15*2)</div> <div class=sectline> (2*2*3) (12*2) (2*15)</div> <div class=sectline> (2*3*2) (2*12) (3*10)</div> <div class=sectline> (3*2*2) (2*2*6) (2*5*3)</div> <div class=sectline> (2*4*3) (3*2*5)</div> <div class=sectline> (2*6*2) (3*5*2)</div> <div class=sectline> (3*2*4) (5*2*3)</div> <div class=sectline> (3*4*2)</div> <div class=sectline> (4*2*3)</div> <div class=sectline> (6*2*2)</div> <div class=sectline> (2*2*2*3)</div> <div class=sectline> (2*2*3*2)</div> <div class=sectline> (2*3*2*2)</div> <div class=sectline> (3*2*2*2)</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];</div> <div class=sectline>whkQ[y_]:=And@@Table[If[EvenQ[m], y[[m]]<=y[[m+1]], y[[m]]>=y[[m+1]]], {m, 1, Length[y]-1}];</div> <div class=sectline>Table[Length[Select[Join@@Permutations/@facs[n], whkQ[#]||whkQ[-#]&]], {n, 100}]</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>The strong version for compositions is <a href="/A025047" title="Number of alternating compositions, i.e., compositions with alternating increases and decreases, starting with either an inc...">A025047</a>, also <a href="/A025048" title="Number of up/down (initially ascending) compositions of n.">A025048</a>, <a href="/A025049" title="Number of down/up (initially descending) compositions of n.">A025049</a>.</div> <div class=sectline>The strong case is <a href="/A348610" title="Number of alternating ordered factorizations of n.">A348610</a>, complement <a href="/A348613" title="Number of non-alternating ordered factorizations of n.">A348613</a>.</div> <div class=sectline>The version for compositions is <a href="/A349052" title="Number of weakly alternating compositions of n.">A349052</a>, complement <a href="/A349053" title="Number of non-weakly alternating integer compositions of n.">A349053</a>.</div> <div class=sectline>As compositions these are ranked by the complement of <a href="/A349057" title="Numbers k such that the k-th composition in standard order is not weakly alternating.">A349057</a>.</div> <div class=sectline><a href="/A001055" title="The multiplicative partition function: number of ways of factoring n with all factors greater than 1 (a(1) = 1 by convention).">A001055</a> counts factorizations, strict <a href="/A045778" title="Number of factorizations of n into distinct factors greater than 1.">A045778</a>, ordered <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="/A001250" title="Number of alternating permutations of order n.">A001250</a> counts alternating permutations, complement <a href="/A348615" title="Number of non-alternating permutations of {1...n}.">A348615</a>.</div> <div class=sectline><a href="/A335434" title="Number of separable factorizations of n into factors > 1.">A335434</a> counts separable factorizations, complement <a href="/A333487" title="Number of inseparable factorizations of n into factors > 1.">A333487</a>.</div> <div class=sectline><a href="/A345164" title="Number of alternating permutations of the multiset of prime factors of n.">A345164</a> counts alternating permutations of prime factors, w/ twins <a href="/A344606" title="Number of alternating permutations of the prime factors of n, counting multiplicity, including twins (x,x).">A344606</a>.</div> <div class=sectline><a href="/A345170" title="Number of integer partitions of n with an alternating permutation.">A345170</a> counts partitions with an alternating permutation.</div> <div class=sectline><a href="/A348379" title="Number of factorizations of n with an alternating permutation.">A348379</a> = factorizations w/ alternating permutation, complement <a href="/A348380" title="Number of factorizations of n without an alternating permutation. Includes all twins (x*x).">A348380</a>.</div> <div class=sectline><a href="/A348611" title="Number of ordered factorizations of n with no adjacent equal factors.">A348611</a> counts anti-run ordered factorizations, complement <a href="/A348616" title="Number of ordered factorizations of n with adjacent equal factors.">A348616</a>.</div> <div class=sectline><a href="/A349060" title="Number of integer partitions of n that are constant or whose part multiplicities, except possibly the first and last, are al...">A349060</a> counts weakly alternating partitions, complement <a href="/A349061" title="Number of integer partitions of n with at least one part of odd multiplicity that is not the first or last.">A349061</a>.</div> <div class=sectline><a href="/A349800" title="Number of integer compositions of n that are weakly alternating and have at least two adjacent equal parts.">A349800</a> = weakly but not strongly alternating compositions, ranked <a href="/A349799" title="Numbers k such that the k-th composition in standard order is weakly alternating but has at least two adjacent equal parts.">A349799</a>.</div> <div class=sectline>Cf. <a href="/A003242" title="Number of compositions of n such that no two adjacent parts are equal (these are sometimes called Carlitz compositions).">A003242</a>, <a href="/A122181" title="Numbers k that can be written as k = x*y*z with 1 < x < y < z (A122180(k) > 0).">A122181</a>, <a href="/A138364" title="The number of Motzkin n-paths with exactly one flat step.">A138364</a>, <a href="/A339846" title="Number of even-length factorizations of n into factors > 1.">A339846</a>, <a href="/A339890" title="Number of odd-length factorizations of n into factors > 1.">A339890</a>, <a href="/A345165" title="Number of integer partitions of n without an alternating permutation.">A345165</a>, <a href="/A345167" title="Numbers k such that the k-th composition in standard order is alternating.">A345167</a>, <a href="/A345194" title="Number of alternating patterns of length n.">A345194</a>, <a href="/A347050" title="Number of factorizations of n that are a twin (x*x) or have an alternating permutation.">A347050</a>, <a href="/A347438" title="Number of unordered factorizations of n with alternating product 1.">A347438</a>, <a href="/A347463" title="Number of ordered factorizations of n with integer alternating product.">A347463</a>, <a href="/A347706" title="Number of factorizations of n that are not a twin (x*x) nor have an alternating permutation.">A347706</a>.</div> <div class=sectline>Sequence in context: <a href="/A296119" title="Number of ways to choose a strict factorization of each factor in a factorization of n.">A296119</a> <a href="/A300836" title="a(n) is the total number of terms (1-digits) in Zeckendorf representation of all proper divisors of n.">A300836</a> <a href="/A118314" title="Erroneous version of A002033.">A118314</a> * <a href="/A002033" title="Number of perfect partitions of n.">A002033</a> <a href="/A074206" title="Kalm谩r's [Kalmar's] problem: number of ordered factorizations of n.">A074206</a> <a href="/A173801" title="The number of primitive numbers k such that 1/k is in the Cantor set and the fraction 1/k has period n.">A173801</a></div> <div class=sectline>Adjacent sequences: <a href="/A349056" title="Number of weakly alternating permutations of the multiset of prime factors of n.">A349056</a> <a href="/A349057" title="Numbers k such that the k-th composition in standard order is not weakly alternating.">A349057</a> <a href="/A349058" title="Number of weakly alternating patterns of length n.">A349058</a> * <a href="/A349060" title="Number of integer partitions of n that are constant or whose part multiplicities, except possibly the first and last, are al...">A349060</a> <a href="/A349061" title="Number of integer partitions of n with at least one part of odd multiplicity that is not the first or last.">A349061</a> <a href="/A349062" title="Powerful numbers (A001694) with a record gap to the next powerful number.">A349062</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>, Dec 04 2021</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 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Contains 382671 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>