keyword:new

<!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>keyword:new - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/search?q=keyword:new\u0026sort=created" 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=%2fsearch%3fq%3dkeyword%3anew%26sort%3dcreated">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="keyword:new - 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="keyword:new" title="Search Query"> <input type=hidden name=sort value="created"> </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> <span dir="LTR">Search:</span> <b>keyword:new</b> <br> <div class=resultbar> <div class=resultline></div> <div> <div class=pagelinkbox> <div> <td>Displaying 1-10 of 497 results found. </div> <div class=pagelinks> <font size=-1>page 1 <a href="/search?q=keyword%3anew&start=10&sort=created">2</a> <a href="/search?q=keyword%3anew&start=20&sort=created">3</a> <a href="/search?q=keyword%3anew&start=30&sort=created">4</a> <a href="/search?q=keyword%3anew&start=40&sort=created">5</a> <a href="/search?q=keyword%3anew&start=50&sort=created">6</a> <a href="/search?q=keyword%3anew&start=60&sort=created">7</a> <a href="/search?q=keyword%3anew&start=70&sort=created">8</a> <a href="/search?q=keyword%3anew&start=80&sort=created">9</a> <a href="/search?q=keyword%3anew&start=90&sort=created">10</a> ... <a href="/search?q=keyword%3anew&start=490&sort=created">50</a> </font> </div> </div> </div> <div>      <font size=-1>Sort: <a href="/search?q=keyword%3anew">relevance</a> | <a href="/search?q=keyword%3anew&sort=references">references</a> | <a href="/search?q=keyword%3anew&sort=number">number</a> | <a href="/search?q=keyword%3anew&sort=modified">modified</a> | created </font>      <font size=-1>Format: long | <a href="/search?q=keyword%3anew&fmt=short&sort=created">short</a> | <a href="/search?q=keyword%3anew&fmt=data&sort=created">data</a> </font> </div> <div class=resultline></div> </div> <div class=space5></div> <div class=sequence> <div class=space1></div> <div class=line></div> <div class=seqhead> <div class=seqnumname> <div class=seqnum> <a href="/A378222">A378222</a> </div> <div class=seqname> Number of ordered factorizations of the odd part of n into factors > 1. </div> </div> <div class=scorerefs> +0<br> 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> </div> <div class=section> <div class=sectname>KEYWORD</div> <div class=sectbody> <div class=sectline><span title="a sequence of nonnegative numbers">nonn</span>,<b><span title="added within the last two weeks">new</span></b></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 class=sequence> <div class=space1></div> <div class=line></div> <div class=seqhead> <div class=seqnumname> <div class=seqnum> <a href="/A377933">A377933</a> </div> <div class=seqname> First differences of consecutive perfect powers m^k with k>=3 (<a href="/A076467" title="Perfect powers m^k where m is a positive integer and k >= 3.">A076467</a>). </div> </div> <div class=scorerefs> +0<br> 0 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>7, 8, 11, 5, 32, 17, 44, 3, 88, 27, 13, 87, 169, 113, 104, 271, 24, 272, 35, 397, 320, 139, 10, 204, 343, 381, 250, 721, 817, 919, 729, 298, 917, 224, 192, 1069, 739, 648, 1519, 1657, 817, 984, 759, 423, 769, 2107, 1053, 1216, 2437, 2611, 1561, 1230, 2977, 3169, 2479, 888</div> <div class=seqdatalinks> (<a href="/A377933/list">list</a>; <a href="/A377933/graph">graph</a>; <a href="/search?q=A377933+-id:A377933">refs</a>; <a href="/A377933/listen">listen</a>; <a href="/history?seq=A377933">history</a>; <a href="/search?q=id:A377933&fmt=text">text</a>; <a href="/A377933/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>1,1</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>Hugo Pfoertner, <a href="/A377933/b377933.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) = <a href="/A076467" title="Perfect powers m^k where m is a positive integer and k >= 3.">A076467</a>(n+1) - <a href="/A076467" title="Perfect powers m^k where m is a positive integer and k >= 3.">A076467</a>(n).</div> </div> </div> <div class=section> <div class=sectname>MAPLE</div> <div class=sectbody> <div class=sectline>N:= 10^5: # for terms <= N of <a href="/A076467" title="Perfect powers m^k where m is a positive integer and k >= 3.">A076467</a></div> <div class=sectline>S:= sort(convert({1, seq(seq(m^k, m = 2 .. floor(N^(1/k))), k=3..ilog2(N))}, list)):</div> <div class=sectline>S[2..-1]-S[1..-2]; # <a href="/wiki/User:Robert_Israel">Robert Israel</a>, Nov 24 2024</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(PARI) lista(nn) = my(S=List(1)); for(x=2, sqrtnint(nn, 3), for(k=3, logint(nn, x), listput(S, x^k))); my(v=Set(S)); vector(#v-1, k, v[k+1]-v[k]); \\ <a href="/wiki/User:Michel_Marcus">Michel Marcus</a>, Nov 24 2024</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A001597" title="Perfect powers: m^k where m > 0 and k >= 2.">A001597</a>, <a href="/A053289" title="First differences of consecutive perfect powers (A001597).">A053289</a>, <a href="/A076467" title="Perfect powers m^k where m is a positive integer and k >= 3.">A076467</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="it is very easy to produce terms of sequence">easy</span>,<b><span title="added within the last two weeks">new</span></b></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:Hugo_Pfoertner">Hugo Pfoertner</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 class=sequence> <div class=space1></div> <div class=line></div> <div class=seqhead> <div class=seqnumname> <div class=seqnum> <a href="/A377934">A377934</a> </div> <div class=seqname> a(n) is the number of perfect powers m^k with k>=3 (<a href="/A076467" title="Perfect powers m^k where m is a positive integer and k >= 3.">A076467</a>) <= 10^n. </div> </div> <div class=scorerefs> +0<br> 0 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>1, 2, 7, 17, 38, 75, 152, 306, 616, 1260, 2598, 5401, 11307, 23798, 50316, 106776, 227236, 484737, 1036002, 2217529, 4752349, 10194727, 21887147, 47020054, 101065880, 217325603, 467484989, 1005881993</div> <div class=seqdatalinks> (<a href="/A377934/list">list</a>; <a href="/A377934/graph">graph</a>; <a href="/search?q=A377934+-id:A377934">refs</a>; <a href="/A377934/listen">listen</a>; <a href="/history?seq=A377934">history</a>; <a href="/search?q=id:A377934&fmt=text">text</a>; <a href="/A377934/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>0,2</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline><a href="/A377934/b377934.txt">Table of n, a(n) for n=0..27.</a></div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>a(0) = 1: 1^k with any k>2 (<= 10^0);</div> <div class=sectline>a(1) = 2: 1 and 2^3 (<=10^1);</div> <div class=sectline>a(2) = 7: 2 powers <= 10 and 16, 27, 32, 64, 81 (<=10^2).</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A070428" title="Number of perfect powers (A001597) not exceeding 10^n.">A070428</a>, <a href="/A076467" title="Perfect powers m^k where m is a positive integer and k >= 3.">A076467</a>, <a href="/A378168" title="a(n) is the number of squares <= 10^n that are not higher powers, i.e., terms of A076467.">A378168</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="more terms are needed! please extend this sequence">more</span>,<b><span title="added within the last two weeks">new</span></b></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:Hugo_Pfoertner">Hugo Pfoertner</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 class=sequence> <div class=space1></div> <div class=line></div> <div class=seqhead> <div class=seqnumname> <div class=seqnum> <a href="/A377867">A377867</a> </div> <div class=seqname> Number of subwords of the form DDDD in nondecreasing Dyck paths of length 2n. </div> </div> <div class=scorerefs> +0<br> 0 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>0, 0, 0, 0, 1, 7, 33, 131, 473, 1608, 5242, 16567, 51123, 154793, 461525, 1358646, 3957088, 11420995, 32707809, 93040751, 263113505, 740238852, 2073098086, 5782387855, 16070206191, 44516728277, 122956408493, 338707969266, 930787894348, 2552224341403, 6984100641117</div> <div class=seqdatalinks> (<a href="/A377867/list">list</a>; <a href="/A377867/graph">graph</a>; <a href="/search?q=A377867+-id:A377867">refs</a>; <a href="/A377867/listen">listen</a>; <a href="/history?seq=A377867">history</a>; <a href="/search?q=id:A377867&fmt=text">text</a>; <a href="/A377867/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>0,6</div> </div> </div> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <div class=sectline>A Dyck path is nondecreasing if the y-coordinates of its valleys form a nondecreasing sequence.</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline><a href="/A377867/b377867.txt">Table of n, a(n) for n=0..30.</a></div> <div class=sectline>E. Barcucci, A. Del Lungo, S. Fezzi, and R. Pinzani, <a href="http://dx.doi.org/10.1016/S0012-365X(97)82778-1">Nondecreasing Dyck paths and q-Fibonacci numbers</a>, Discrete Math., 170 (1997), 211-217.</div> <div class=sectline>脡va Czabarka, Rigoberto Fl贸rez, Leandro Junes and Jos茅 L. Ram铆rez, <a href="https://doi.org/10.1016/j.disc.2018.06.032">Enumerations of peaks and valleys on non-decreasing Dyck paths</a>, Discrete Math., Vol. 341, No. 10 (2018), pp. 2789-2807. See p. 2798.</div> <div class=sectline>Rigoberto Fl贸rez, Leandro Junes, and Jos茅 L. Ram铆rez, <a href="https://doi.org/10.1016/j.disc.2019.06.018">Enumerating several aspects of non-decreasing Dyck paths</a>, Discrete Mathematics, Vol. 342, Issue 11 (2019), 3079-3097. See page 3092.</div> <div class=sectline><a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (10,-39,74,-69,28,-4).</div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>a(n) = (3*(n-2)*L(2*n-4) - 3*F(2*n+1))/5 + (n+9)*2^(n-4) for n>=3, where F(n) = <a href="/A000045" title="Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.">A000045</a>(n) and L(n) = <a href="/A000032" title="Lucas numbers beginning at 2: L(n) = L(n-1) + L(n-2), L(0) = 2, L(1) = 1.">A000032</a>(n).</div> <div class=sectline>G.f.: x^4*(1 - 3*x + 2*x^2 + x^4)/((1 - 2*x)^2*(1 - 3*x + x^2)^2).</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>Table[If[n < 3, 0, (3*(n-2)*LucasL[2*n-4]-3*Fibonacci[2*n+1])/5+(n+9)*2^(n-4)], {n, 0, 20}]</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A000032" title="Lucas numbers beginning at 2: L(n) = L(n-1) + L(n-2), L(0) = 2, L(1) = 1.">A000032</a>, <a href="/A000045" title="Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.">A000045</a>, <a href="/A377679" title="Number of subwords of the form DDD in nondecreasing Dyck paths of length 2n.">A377679</a>, <a href="/A377670" title="Number of subwords of the form UDD in nondecreasing Dyck paths of length 2n.">A377670</a>, <a href="/A375995" title="Number of subwords of the form UUUU in nondecreasing Dyck paths of length 2n.">A375995</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="it is very easy to produce terms of sequence">easy</span>,<b><span title="added within the last two weeks">new</span></b></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:Rigoberto_Florez">Rigoberto Florez</a>, Nov 10 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 class=sequence> <div class=space1></div> <div class=line></div> <div class=seqhead> <div class=seqnumname> <div class=seqnum> <a href="/A377866">A377866</a> </div> <div class=seqname> Number of subwords of the form DUUD or DDUUD in nondecreasing Dyck paths of length 2n. </div> </div> <div class=scorerefs> +0<br> 0 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>0, 0, 0, 1, 5, 18, 59, 185, 564, 1685, 4957, 14406, 41455, 118321, 335400, 945193, 2650229, 7398330, 20573219, 57013865, 157517532, 433993661, 1192779085, 3270835566, 8950887895, 24448816993, 66665369424, 181489721425, 493361278949</div> <div class=seqdatalinks> (<a href="/A377866/list">list</a>; <a href="/A377866/graph">graph</a>; <a href="/search?q=A377866+-id:A377866">refs</a>; <a href="/A377866/listen">listen</a>; <a href="/history?seq=A377866">history</a>; <a href="/search?q=id:A377866&fmt=text">text</a>; <a href="/A377866/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>0,5</div> </div> </div> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <div class=sectline>A Dyck path is nondecreasing if the y-coordinates of its valleys form a nondecreasing sequence.</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline><a href="/A377866/b377866.txt">Table of n, a(n) for n=0..28.</a></div> <div class=sectline>E. Barcucci, A. Del Lungo, S. Fezzi, and R. Pinzani, <a href="http://dx.doi.org/10.1016/S0012-365X(97)82778-1">Nondecreasing Dyck paths and q-Fibonacci numbers</a>, Discrete Math., 170 (1997), 211-217.</div> <div class=sectline>脡va Czabarka, Rigoberto Fl贸rez, Leandro Junes and Jos茅 L. Ram铆rez, <a href="https://doi.org/10.1016/j.disc.2018.06.032">Enumerations of peaks and valleys on non-decreasing Dyck paths</a>, Discrete Math., Vol. 341, No. 10 (2018), pp. 2789-2807. See p. 2798.</div> <div class=sectline>Rigoberto Fl贸rez, Leandro Junes, and Jos茅 L. Ram铆rez, <a href="https://doi.org/10.1016/j.disc.2019.06.018">Enumerating several aspects of non-decreasing Dyck paths</a>, Discrete Mathematics, Vol. 342, Issue 11 (2019), 3079-3097. See page 3092.</div> <div class=sectline><a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (6,-11,6,-1).</div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>a(n) = (2*n*L(2*n-5) - 6*F(2*n-6) - F(2*n-7))/5 for n>=3, where F(n)=<a href="/A000045" title="Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.">A000045</a>(n) and L(n)=<a href="/A000032" title="Lucas numbers beginning at 2: L(n) = L(n-1) + L(n-2), L(0) = 2, L(1) = 1.">A000032</a>(n).</div> <div class=sectline>G.f.: -x^3*(x^2+x-1)/ (x^2-3*x+1)^2.</div> <div class=sectline>E.g.f.: exp(3*x/2)*(5*(35 - 8x)*cosh(sqrt(5)*x/2) - sqrt(5)*(79 - 20*x)*sinh(sqrt(5)*x/2))/25 - 7 - x. - <a href="/wiki/User:Stefano_Spezia">Stefano Spezia</a>, Nov 10 2024</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>Table[If[n<3, 0, (2*n*LucasL[2*n-5]-6*Fibonacci[2*n-6]-Fibonacci[2*n-7])/5], {n, 0, 20}]</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A000032" title="Lucas numbers beginning at 2: L(n) = L(n-1) + L(n-2), L(0) = 2, L(1) = 1.">A000032</a>, <a href="/A000045" title="Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.">A000045</a>, <a href="/A377679" title="Number of subwords of the form DDD in nondecreasing Dyck paths of length 2n.">A377679</a>, <a href="/A377670" title="Number of subwords of the form UDD in nondecreasing Dyck paths of length 2n.">A377670</a>, <a href="/A375995" title="Number of subwords of the form UUUU in nondecreasing Dyck paths of length 2n.">A375995</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="it is very easy to produce terms of sequence">easy</span>,<b><span title="added within the last two weeks">new</span></b></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:Rigoberto_Florez">Rigoberto Florez</a>, Nov 10 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 class=sequence> <div class=space1></div> <div class=line></div> <div class=seqhead> <div class=seqnumname> <div class=seqnum> <a href="/A377857">A377857</a> </div> <div class=seqname> Number of subwords of the form UUUD in nondecreasing Dyck paths of length 2n. </div> </div> <div class=scorerefs> +0<br> 0 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>0, 0, 0, 1, 5, 18, 60, 191, 589, 1775, 5257, 15360, 44394, 127171, 361595, 1021693, 2871245, 8031246, 22372344, 62096135, 171797257, 473928875, 1304007889, 3579517116, 9804791910, 26804181643, 73145473655, 199276078201, 542076556949, 1472491141770, 3994615719732</div> <div class=seqdatalinks> (<a href="/A377857/list">list</a>; <a href="/A377857/graph">graph</a>; <a href="/search?q=A377857+-id:A377857">refs</a>; <a href="/A377857/listen">listen</a>; <a href="/history?seq=A377857">history</a>; <a href="/search?q=id:A377857&fmt=text">text</a>; <a href="/A377857/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>0,5</div> </div> </div> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <div class=sectline>A Dyck path is nondecreasing if the y-coordinates of its valleys form a nondecreasing sequence.</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline><a href="/A377857/b377857.txt">Table of n, a(n) for n=0..30.</a></div> <div class=sectline>E. Barcucci, A. Del Lungo, S. Fezzi, and R. Pinzani, <a href="http://dx.doi.org/10.1016/S0012-365X(97)82778-1">Nondecreasing Dyck paths and q-Fibonacci numbers</a>, Discrete Math., 170 (1997), 211-217.</div> <div class=sectline>脡va Czabarka, Rigoberto Fl贸rez, Leandro Junes and Jos茅 L. Ram铆rez, <a href="https://doi.org/10.1016/j.disc.2018.06.032">Enumerations of peaks and valleys on non-decreasing Dyck paths</a>, Discrete Math., Vol. 341, No. 10 (2018), pp. 2789-2807. See p. 2798.</div> <div class=sectline>Rigoberto Fl贸rez, Leandro Junes, and Jos茅 L. Ram铆rez, <a href="https://doi.org/10.1016/j.disc.2019.06.018">Enumerating several aspects of non-decreasing Dyck paths</a>, Discrete Mathematics, Vol. 342, Issue 11 (2019), 3079-3097. See page 3092.</div> <div class=sectline><a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (6,-11,6,-1).</div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>a(n) = n*F(2*n-5) - L(2*n-6) for n>=3, where F(n) = <a href="/A000045" title="Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.">A000045</a>(n) and L(n) = <a href="/A000032" title="Lucas numbers beginning at 2: L(n) = L(n-1) + L(n-2), L(0) = 2, L(1) = 1.">A000032</a>(n).</div> <div class=sectline>G.f.: x^3*(1 - x)^2*(1 + x)/(1 - 3*x + x^2)^2.</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>Table[If[n<3, 0, n Fibonacci[2n-5]-LucasL[2n-6]], {n, 0, 30}]</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A000032" title="Lucas numbers beginning at 2: L(n) = L(n-1) + L(n-2), L(0) = 2, L(1) = 1.">A000032</a>, <a href="/A000045" title="Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.">A000045</a>, <a href="/A377679" title="Number of subwords of the form DDD in nondecreasing Dyck paths of length 2n.">A377679</a>, <a href="/A377670" title="Number of subwords of the form UDD in nondecreasing Dyck paths of length 2n.">A377670</a>, <a href="/A375995" title="Number of subwords of the form UUUU in nondecreasing Dyck paths of length 2n.">A375995</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="it is very easy to produce terms of sequence">easy</span>,<b><span title="added within the last two weeks">new</span></b></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:Rigoberto_Florez">Rigoberto Florez</a>, Nov 09 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 class=sequence> <div class=space1></div> <div class=line></div> <div class=seqhead> <div class=seqnumname> <div class=seqnum> <a href="/A377280">A377280</a> </div> <div class=seqname> Given n cards, each time you reversing the order of the top 1, 2, 3, ..., n-1, n cards, then repeat reversing 1, 2, 3, ... cards. Do reversing at least once. the minimum number of steps required to return all the cards to their original position. </div> </div> <div class=scorerefs> +0<br> 0 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>1, 4, 9, 12, 25, 36, 28, 32, 81, 60, 121, 120, 117, 196, 75, 80, 204, 324, 228, 200, 147, 264, 529, 504, 200, 676, 540, 252, 841, 900, 186, 192, 1089, 748, 1225, 324, 740, 1140, 1521, 1080, 1681, 336, 1204, 484, 540, 460, 1692, 1152, 735, 2500, 2601, 624, 2809, 972, 1980, 784, 2508, 696, 1416, 3300</div> <div class=seqdatalinks> (<a href="/A377280/list">list</a>; <a href="/A377280/graph">graph</a>; <a href="/search?q=A377280+-id:A377280">refs</a>; <a href="/A377280/listen">listen</a>; <a href="/history?seq=A377280">history</a>; <a href="/search?q=id:A377280&fmt=text">text</a>; <a href="/A377280/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>1,2</div> </div> </div> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <div class=sectline>The sequence is only restored after one of the full length n reversal, so a(n) >= n.</div> <div class=sectline>The process of reversing blocks from 1 to n corresponds to the order transformation of numbers in sequence <a href="/A130517" title="Triangle read by rows: row n counts down from n in steps of 2, then counts up the remaining elements in the set {1,2,...,n},...">A130517</a>.</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline><a href="/A377280/b377280.txt">Table of n, a(n) for n=1..60.</a></div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>a(n) = n * <a href="/A003558" title="Least number m > 0 such that 2^m == +-1 (mod 2n + 1).">A003558</a>(n).</div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>For example, using "abc" to represent three cards, the card positions at the end of each step are: abc, bac, cab, cab, acb, bca, bca, cba, abc. Therefore, it takes 9 steps. If there are 4 cards "abcd", the sequence of changes is: abcd, bacd, cabd, dbac, dbac, bdac, adbc, cbda, cbda, bcda, dcba, abcd, so it takes 12 steps</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(PARI) <a href="/A377280" title="Given n cards, each time you reversing the order of the top 1, 2, 3, ..., n-1, n cards, then repeat reversing 1, 2, 3, ... c...">A377280</a>(n)=my(M=Mod(2, 2*n+1), o=znorder(M)); if(o%2==0&&M^(o/2)==-1, n*o/2, o*n) \\ <a href="/wiki/User:Kevin_Ryde">Kevin Ryde</a></div> <div class=sectline>(Python)</div> <div class=sectline>from sympy.ntheory import n_order</div> <div class=sectline>def <a href="/A377280" title="Given n cards, each time you reversing the order of the top 1, 2, 3, ..., n-1, n cards, then repeat reversing 1, 2, 3, ... c...">A377280</a>(n):</div> <div class=sectline> modular = 2*n + 1</div> <div class=sectline> order = n_order(2, 2*n+1)</div> <div class=sectline> if order % 2 == 0 and pow(2, order//2, modular) == modular - 1:</div> <div class=sectline> return (order//2) * n</div> <div class=sectline> else:</div> <div class=sectline> return order * n # after <a href="/wiki/User:Kevin_Ryde">Kevin Ryde</a></div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A003558" title="Least number m > 0 such that 2^m == +-1 (mod 2n + 1).">A003558</a>. <a href="/A130517" title="Triangle read by rows: row n counts down from n in steps of 2, then counts up the remaining elements in the set {1,2,...,n},...">A130517</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>,<b><span title="added within the last two weeks">new</span></b></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:Youhua_Li">Youhua Li</a>, Oct 22 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 class=sequence> <div class=space1></div> <div class=line></div> <div class=seqhead> <div class=seqnumname> <div class=seqnum> <a href="/A377757">A377757</a> </div> <div class=seqname> Number of possibilities to place "hat" monotiles onto the first n hexagons in an counterclockwise order such that at least one more ring of tiles can be placed around it. </div> </div> <div class=scorerefs> +0<br> 0 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>1, 3, 4, 7, 18, 20, 20, 29, 30, 30, 31, 37, 37, 39, 39, 42, 45, 48, 48</div> <div class=seqdatalinks> (<a href="/A377757/list">list</a>; <a href="/A377757/graph">graph</a>; <a href="/search?q=A377757+-id:A377757">refs</a>; <a href="/A377757/listen">listen</a>; <a href="/history?seq=A377757">history</a>; <a href="/search?q=id:A377757&fmt=text">text</a>; <a href="/A377757/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>1,2</div> </div> </div> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <div class=sectline>Fig. 1.1 in the paper "An aperiodic monotile" by Smith et al. (2023) shows an example of tiling the plane with "hat" monotiles. A "hat" monotile covers two thirds of a hexagon and one third of two neighboring hexagons, respectively. A hexagon is assigned a tile if the tile covers two thirds of the hexagon. Since the surface of a "hat" tile is 4/3 the surface of a hexagon, every forth hexagon on average is covered by parts of three different tiles. In this case we assign a zero to this hexagon.</div> <div class=sectline>We assign the number "1" to a tile that has the orientation of the dark grey tile in Fig. 1.1 of the paper by Smith. Tiles can only be rotated by multiples of 60 deg and for each counterclockwise rotation of the tile by 60 deg, we increase the count by 1 such that unreflected tiles are assigned the numbers from 1 to 6. Reflected tiles are assigned accordingly the numbers from 7 to 12.</div> <div class=sectline>Without loss of generality we place a tile "1" in the central hexagon like in the quoted Fig 1.1 (any other tiling can be transformed into this tiling by rotation or reflection). As next hexagon to be filled we choose the hexagon below (direction "6 o'clock"), because this is the only hexagon that must be assigned an unreflected tile with number 1 to 6, all other five neighboring hexagons may be a assigned a zero or a reflected tile. The tiling continues through the surrounding hexagons in counterclockwise direction ("4, 2, 12, 10 and 8 o'clock"), before the next ring is started again at "6 o'clock". Using this terminology the tiling in Fig. 1.1 is described by 1,4,0,2,7,6,0,...</div> <div class=sectline>a(n) is the number of possible assignments of the first n hexagons such that at least one more ring of tiles can be placed around it. It would be desirable to demand that the tiling can be continued to infinity, but this is much more difficult to prove.</div> <div class=sectline>An open question is: What is a(n)/n when n tends to infinity?</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline><a href="/A377757/b377757.txt">Table of n, a(n) for n=1..19.</a></div> <div class=sectline>David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss, <a href="https://arxiv.org/abs/2303.10798">An aperiodic monotile</a>, arXiv:2303.10798 [math.CO], 2023.</div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>Example: a(3)=4: 1,1,0; 1,4,0; 1,4,6; and 1,5,12 are the 4 tiling options for the first three hexagons such that further tiling is possible. a(7)=20: there are 20 different ways to tile a ring around the central hexagon.</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A342285" title="Coordination sequence with respect to the central vertex of a dodecagon-based tiling of the plane by copies of a certain Gol...">A342285</a>, <a href="/A363348" title="Turn sequence of a non-Eulerian path for drawing an infinite aperiodic tiling based on the "hat" monotile. See the comments ...">A363348</a>, <a href="/A363445" title="Turn sequence of a fractal-like curve which is also the perimeter around an aperiodic tiling based on the "hat" monotile. Se...">A363445</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="more terms are needed! please extend this sequence">more</span>,<b><span title="added within the last two weeks">new</span></b></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:Ruediger_Jehn">Ruediger Jehn</a> and <a href="/wiki/User:Kester_Habermann">Kester Habermann</a>, Nov 06 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 class=sequence> <div class=space1></div> <div class=line></div> <div class=seqhead> <div class=seqnumname> <div class=seqnum> <a href="/A377766">A377766</a> </div> <div class=seqname> Even numbers whose sum of proper (or aliquot) divisors is a prime. </div> </div> <div class=scorerefs> +0<br> 0 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>4, 8, 32, 50, 98, 128, 242, 324, 338, 392, 722, 784, 800, 1058, 1250, 1444, 2304, 2312, 2450, 2704, 2738, 3600, 3872, 5408, 5476, 5618, 6272, 6728, 7442, 7688, 8192, 9248, 11552, 12482, 12800, 14400, 14884, 15488, 15842, 16562, 16900, 16928, 17672, 18050, 19208, 21632, 21904, 22500, 23762, 25088</div> <div class=seqdatalinks> (<a href="/A377766/list">list</a>; <a href="/A377766/graph">graph</a>; <a href="/search?q=A377766+-id:A377766">refs</a>; <a href="/A377766/listen">listen</a>; <a href="/history?seq=A377766">history</a>; <a href="/search?q=id:A377766&fmt=text">text</a>; <a href="/A377766/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>1,1</div> </div> </div> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <div class=sectline>Even terms of <a href="/A037020" title="Numbers whose sum of proper (or aliquot) divisors is a prime.">A037020</a>. Numbers from <a href="/A088827" title="Even numbers with odd abundance: even squares or two times squares.">A088827</a> (2n^2 or 4n^2) are the only aliquot sum transition from even to odd.</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline><a href="/A377766/b377766.txt">Table of n, a(n) for n=1..50.</a></div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>The aliquot divisors of 32 are 1, 2, 4, 8 and 16, whose sum is 31, a prime, so 32 is a term.</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>Select[2Range[13000], PrimeQ[DivisorSigma[1, #]-#] &] (* <a href="/wiki/User:Stefano_Spezia">Stefano Spezia</a>, Nov 08 2024 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(PARI) is_a377766(n) = !(n%2) && isprime(sigma(n)-n) \\ <a href="/wiki/User:Hugo_Pfoertner">Hugo Pfoertner</a>, Nov 07 2024</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Intersection of <a href="/A005843" title="The nonnegative even numbers: a(n) = 2n.">A005843</a> and <a href="/A037020" title="Numbers whose sum of proper (or aliquot) divisors is a prime.">A037020</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="it is very easy to produce terms of sequence">easy</span>,<b><span title="added within the last two weeks">new</span></b></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:Ophir_Spector">Ophir Spector</a>, Nov 06 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 class=sequence> <div class=space1></div> <div class=line></div> <div class=seqhead> <div class=seqnumname> <div class=seqnum> <a href="/A377643">A377643</a> </div> <div class=seqname> a(n) is the number of terms in the trajectory when the map x -> 2+sopfr(x) is iterated, starting from x = n until x = 8, with sopfr = <a href="/A001414" title="Integer log of n: sum of primes dividing n (with repetition). Also called sopfr(n).">A001414</a>. </div> </div> <div class=scorerefs> +0<br> 0 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>7, 6, 5, 5, 4, 4, 3, 1, 2, 3, 6, 3, 5, 7, 4, 4, 6, 4, 5, 7, 4, 5, 5, 7, 4, 7, 7, 6, 7, 4, 6, 4, 5, 5, 8, 4, 6, 6, 5, 6, 8, 8, 7, 7, 6, 8, 6, 6, 5, 8, 6, 6, 6, 6, 5, 5, 8, 6, 7, 8, 6, 9, 5, 8, 8, 5, 8, 6, 7, 5, 7, 8, 6, 9, 5, 5, 8, 8, 9, 5, 8, 7, 9, 5, 8, 7, 6, 6, 7, 5, 6, 8, 5, 7, 8, 5, 7, 5, 6, 5</div> <div class=seqdatalinks> (<a href="/A377643/list">list</a>; <a href="/A377643/graph">graph</a>; <a href="/search?q=A377643+-id:A377643">refs</a>; <a href="/A377643/listen">listen</a>; <a href="/history?seq=A377643">history</a>; <a href="/search?q=id:A377643&fmt=text">text</a>; <a href="/A377643/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>1,1</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline><a href="/A377643/b377643.txt">Table of n, a(n) for n=1..100.</a></div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>For n=1, the trajectory from n down to 8 comprises a(1) = 7 terms: 1 -> 2 -> 4 -> 6 -> 7 -> 9 -> 8.</div> </div> </div> <div class=section> <div class=sectname>MAPLE</div> <div class=sectbody> <div class=sectline>f := proc(n)</div> <div class=sectline> add(op(1, i) * op(2, i), i = ifactors(n)[2]):</div> <div class=sectline>end proc:</div> <div class=sectline>g := proc(n)</div> <div class=sectline> 2 + f(n):</div> <div class=sectline>end proc:</div> <div class=sectline><a href="/A377643" title="a(n) is the number of terms in the trajectory when the map x -> 2+sopfr(x) is iterated, starting from x = n until x = 8, wit...">A377643</a> := proc(n)</div> <div class=sectline>local k, result:</div> <div class=sectline> k := 1:</div> <div class=sectline>result := n:</div> <div class=sectline>while result <> 8 do</div> <div class=sectline>result := g(result):</div> <div class=sectline>k := k + 1:</div> <div class=sectline>end do:</div> <div class=sectline>k:</div> <div class=sectline>end proc:</div> <div class=sectline><a href="/A377643" title="a(n) is the number of terms in the trajectory when the map x -> 2+sopfr(x) is iterated, starting from x = n until x = 8, wit...">A377643</a>(8) := 1:</div> <div class=sectline>map(<a href="/A377643" title="a(n) is the number of terms in the trajectory when the map x -> 2+sopfr(x) is iterated, starting from x = n until x = 8, wit...">A377643</a>, [$1..100]);</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>s[n_] := 2 + Plus @@ Times @@@ FactorInteger[n]; s[1] = 2; a[n_] := Length@ NestWhileList[s, n, # != 8 &]; Array[a, 100] (* <a href="/wiki/User:Amiram_Eldar">Amiram Eldar</a>, Nov 07 2024 *)</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A001414" title="Integer log of n: sum of primes dividing n (with repetition). Also called sopfr(n).">A001414</a> (sopfr).</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>,<b><span title="added within the last two weeks">new</span></b></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:Rafik_Khalfi">Rafik Khalfi</a>, Nov 03 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 class=resultbar> <div class=resultline></div> <div> <div class=pagelinkbox> <div></div> <div class=pagelinks> <font size=-1>page 1 <a href="/search?q=keyword%3anew&start=10&sort=created">2</a> <a href="/search?q=keyword%3anew&start=20&sort=created">3</a> <a href="/search?q=keyword%3anew&start=30&sort=created">4</a> <a href="/search?q=keyword%3anew&start=40&sort=created">5</a> <a href="/search?q=keyword%3anew&start=50&sort=created">6</a> <a href="/search?q=keyword%3anew&start=60&sort=created">7</a> <a href="/search?q=keyword%3anew&start=70&sort=created">8</a> <a href="/search?q=keyword%3anew&start=80&sort=created">9</a> <a href="/search?q=keyword%3anew&start=90&sort=created">10</a> ... <a href="/search?q=keyword%3anew&start=490&sort=created">50</a> </font> </div> </div> </div> <div class=resultline></div> </div> <center><p><font size=-1> Search completed in 0.142 seconds </font></p></center> </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 14:08 EST 2024. 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