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 501 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=500&sort=created">51</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="/A382624">A382624</a> </div> <div class=seqname> Positive integers such that every even digit except the leftmost is immediately preceded by a larger digit. </div> </div> <div class=scorerefs> +0<br> 0 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 17, 19, 20, 21, 23, 25, 27, 29, 30, 31, 32, 33, 35, 37, 39, 40, 41, 42, 43, 45, 47, 49, 50, 51, 52, 53, 54, 55, 57, 59, 60, 61, 62, 63, 64, 65, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 84, 85, 86, 87, 89, 90</div> <div class=seqdatalinks> (<a href="/A382624/list">list</a>; <a href="/A382624/graph">graph</a>; <a href="/search?q=A382624+-id:A382624">refs</a>; <a href="/A382624/listen">listen</a>; <a href="/history?seq=A382624">history</a>; <a href="/search?q=id:A382624&fmt=text">text</a>; <a href="/A382624/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>Conjecture: these are the terms of <a href="/A382621" title="Lexicographically earliest sequence of distinct positive integers such that if a digit d in the digit stream (ignoring comma...">A382621</a>, sorted.</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>Paolo Xausa, <a href="/A382624/b382624.txt">Table of n, a(n) for n = 1..10000</a></div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>A382624Q[k_] := FreeQ[Partition[IntegerDigits[k], 2, 1], {i_, j_?EvenQ} /; i <= j];</div> <div class=sectline>Select[Range[100], A382624Q]</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(Python)</div> <div class=sectline>def ok(n):</div> <div class=sectline> s = str(n)</div> <div class=sectline> return all(s[i+1] < s[i] for i in range(len(s)-1) if s[i+1] in "02468")</div> <div class=sectline>print([k for k in range(1, 91) if ok(k)]) # <a href="/wiki/User:Michael_S._Branicky">Michael S. Branicky</a>, Apr 03 2025</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A377912" title="Numbers such that every even digit except the last is immediately followed by a strictly larger digit.">A377912</a>, <a href="/A382465" title="Positive integers such that every even digit except the first is immediately preceded by a smaller digit.">A382465</a>, <a href="/A382621" title="Lexicographically earliest sequence of distinct positive integers such that if a digit d in the digit stream (ignoring comma...">A382621</a>, <a href="/A382623" title="Positive integers that contain an even digit d immediately preceded by a digit <= d.">A382623</a> (complement).</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="dependent on base used for sequence">base</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:Paolo_Xausa">Paolo Xausa</a>, Apr 01 2025</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="/A382686">A382686</a> </div> <div class=seqname> Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa <a href="/A372272" title="Decimal expansion of the middle positive zero of the Legendre polynomial of degree 6.">A372272</a>. </div> </div> <div class=scorerefs> +0<br> 0 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>3, 6, 0, 7, 6, 1, 5, 7, 3, 0, 4, 8, 1, 3, 8, 6, 0, 7, 5, 6, 9, 8, 3, 3, 5, 1, 3, 8, 3, 7, 7, 1, 6, 1, 1, 1, 6, 6, 1, 5, 2, 1, 8, 9, 2, 7, 4, 6, 7, 4, 5, 4, 8, 2, 2, 8, 9, 7, 3, 9, 2, 4, 0, 2, 3, 7, 1, 4, 0, 0, 3, 7, 8, 3, 7, 2, 6, 1, 7, 1, 8, 3, 2, 0, 9, 6, 2</div> <div class=seqdatalinks> (<a href="/A382686/list">list</a>; <a href="/A382686/constant">constant</a>; <a href="/A382686/graph">graph</a>; <a href="/search?q=A382686+-id:A382686">refs</a>; <a href="/A382686/listen">listen</a>; <a href="/history?seq=A382686">history</a>; <a href="/search?q=id:A382686&fmt=text">text</a>; <a href="/A382686/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>0,1</div> </div> </div> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <div class=sectline>There are floor(k/2) positive zeros of the Legendre polynomial of degree k:</div> <div class=sectline> k | zeros | corresponding weights</div> <div class=sectline>---+---------------------------+--------------------------</div> <div class=sectline> 2 | <a href="/A020760" title="Decimal expansion of 1/sqrt(3).">A020760</a> | <a href="/A000007" title="The characteristic function of {0}: a(n) = 0^n.">A000007</a>*10</div> <div class=sectline> 3 | <a href="/A010513" title="Decimal expansion of square root of 60.">A010513</a>/10 | <a href="/A010716" title="Constant sequence: the all 5's sequence.">A010716</a></div> <div class=sectline> 4 | <a href="/A372267" title="Decimal expansion of the smallest positive zero of the Legendre polynomial of degree 4.">A372267</a>, <a href="/A372268" title="Decimal expansion of the largest positive zero of the Legendre polynomial of degree 4.">A372268</a> | <a href="/A382103" title="Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372267.">A382103</a>, <a href="/A382104" title="Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372268.">A382104</a></div> <div class=sectline> 5 | <a href="/A372269" title="Decimal expansion of the smallest positive zero of the Legendre polynomial of degree 5.">A372269</a>, <a href="/A372270" title="Decimal expansion of the largest positive zero of the Legendre polynomial of degree 5.">A372270</a> | <a href="/A382105" title="Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372269.">A382105</a>, <a href="/A382106" title="Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372270.">A382106</a></div> <div class=sectline> 6 | <a href="/A372271" title="Decimal expansion of the smallest positive zero of the Legendre polynomial of degree 6.">A372271</a>, <a href="/A372272" title="Decimal expansion of the middle positive zero of the Legendre polynomial of degree 6.">A372272</a>, <a href="/A372273" title="Decimal expansion of the largest positive zero of the Legendre polynomial of degree 6.">A372273</a> | <a href="/A382107" title="Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372271.">A382107</a>, this sequence, <a href="/A382687" title="Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372273.">A382687</a></div> <div class=sectline> 7 | <a href="/A372274" title="Decimal expansion of the smallest positive zero of the Legendre polynomial of degree 7.">A372274</a>, <a href="/A372275" title="Decimal expansion of the middle positive zero of the Legendre polynomial of degree 7.">A372275</a>, <a href="/A372276" title="Decimal expansion of the largest positive zero of the Legendre polynomial of degree 7.">A372276</a> | <a href="/A382688" title="Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372274.">A382688</a>, <a href="/A382689" title="Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372275.">A382689</a>, <a href="/A382690" title="Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372276.">A382690</a></div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline><a href="/A382686/b382686.txt">Table of n, a(n) for n=0..86.</a></div> <div class=sectline>A.H.M. Smeets, <a href="/A382103/a382103.py.txt">Python program for Legendre-Gauss quadrature constants.</a></div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Legendre-GaussQuadrature.html">Legendre-Gauss Quadrature</a>.</div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>0.36076157304813860756983351383771611166152189274674...</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A372272" title="Decimal expansion of the middle positive zero of the Legendre polynomial of degree 6.">A372272</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="a decimal expansion of a number"><a href="/A382686/constant">cons</a></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:A.H.M._Smeets">A.H.M. Smeets</a>, Apr 03 2025</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="/A382687">A382687</a> </div> <div class=seqname> Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa <a href="/A372273" title="Decimal expansion of the largest positive zero of the Legendre polynomial of degree 6.">A372273</a>. </div> </div> <div class=scorerefs> +0<br> 0 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>1, 7, 1, 3, 2, 4, 4, 9, 2, 3, 7, 9, 1, 7, 0, 3, 4, 5, 0, 4, 0, 2, 9, 6, 1, 4, 2, 1, 7, 2, 7, 3, 2, 8, 9, 3, 5, 2, 6, 8, 2, 2, 5, 0, 1, 4, 8, 4, 0, 4, 3, 9, 8, 2, 3, 9, 8, 6, 3, 5, 4, 3, 9, 7, 9, 8, 9, 4, 5, 7, 6, 0, 5, 4, 2, 3, 4, 0, 1, 5, 4, 6, 4, 7, 9, 2, 7</div> <div class=seqdatalinks> (<a href="/A382687/list">list</a>; <a href="/A382687/constant">constant</a>; <a href="/A382687/graph">graph</a>; <a href="/search?q=A382687+-id:A382687">refs</a>; <a href="/A382687/listen">listen</a>; <a href="/history?seq=A382687">history</a>; <a href="/search?q=id:A382687&fmt=text">text</a>; <a href="/A382687/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>COMMENTS</div> <div class=sectbody> <div class=sectline>There are floor(k/2) positive zeros of the Legendre polynomial of degree k:</div> <div class=sectline> k | zeros | corresponding weights</div> <div class=sectline>---+---------------------------+--------------------------</div> <div class=sectline> 2 | <a href="/A020760" title="Decimal expansion of 1/sqrt(3).">A020760</a> | <a href="/A000007" title="The characteristic function of {0}: a(n) = 0^n.">A000007</a>*10</div> <div class=sectline> 3 | <a href="/A010513" title="Decimal expansion of square root of 60.">A010513</a>/10 | <a href="/A010716" title="Constant sequence: the all 5's sequence.">A010716</a></div> <div class=sectline> 4 | <a href="/A372267" title="Decimal expansion of the smallest positive zero of the Legendre polynomial of degree 4.">A372267</a>, <a href="/A372268" title="Decimal expansion of the largest positive zero of the Legendre polynomial of degree 4.">A372268</a> | <a href="/A382103" title="Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372267.">A382103</a>, <a href="/A382104" title="Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372268.">A382104</a></div> <div class=sectline> 5 | <a href="/A372269" title="Decimal expansion of the smallest positive zero of the Legendre polynomial of degree 5.">A372269</a>, <a href="/A372270" title="Decimal expansion of the largest positive zero of the Legendre polynomial of degree 5.">A372270</a> | <a href="/A382105" title="Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372269.">A382105</a>, <a href="/A382106" title="Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372270.">A382106</a></div> <div class=sectline> 6 | <a href="/A372271" title="Decimal expansion of the smallest positive zero of the Legendre polynomial of degree 6.">A372271</a>, <a href="/A372272" title="Decimal expansion of the middle positive zero of the Legendre polynomial of degree 6.">A372272</a>, <a href="/A372273" title="Decimal expansion of the largest positive zero of the Legendre polynomial of degree 6.">A372273</a> | <a href="/A382107" title="Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372271.">A382107</a>, <a href="/A382686" title="Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372272.">A382686</a>, this sequence</div> <div class=sectline> 7 | <a href="/A372274" title="Decimal expansion of the smallest positive zero of the Legendre polynomial of degree 7.">A372274</a>, <a href="/A372275" title="Decimal expansion of the middle positive zero of the Legendre polynomial of degree 7.">A372275</a>, <a href="/A372276" title="Decimal expansion of the largest positive zero of the Legendre polynomial of degree 7.">A372276</a> | <a href="/A382688" title="Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372274.">A382688</a>, <a href="/A382689" title="Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372275.">A382689</a>, <a href="/A382690" title="Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372276.">A382690</a></div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline><a href="/A382687/b382687.txt">Table of n, a(n) for n=0..86.</a></div> <div class=sectline>A.H.M. Smeets, <a href="/A382103/a382103.py.txt">Python program for Legendre-Gauss quadrature constants.</a></div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Legendre-GaussQuadrature.html">Legendre-Gauss Quadrature</a>.</div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>0.17132449237917034504029614217273289352682250148404...</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A372273" title="Decimal expansion of the largest positive zero of the Legendre polynomial of degree 6.">A372273</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="a decimal expansion of a number"><a href="/A382687/constant">cons</a></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:A.H.M._Smeets">A.H.M. Smeets</a>, Apr 03 2025</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="/A382688">A382688</a> </div> <div class=seqname> Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa <a href="/A372274" title="Decimal expansion of the smallest positive zero of the Legendre polynomial of degree 7.">A372274</a>. </div> </div> <div class=scorerefs> +0<br> 0 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>3, 8, 1, 8, 3, 0, 0, 5, 0, 5, 0, 5, 1, 1, 8, 9, 4, 4, 9, 5, 0, 3, 6, 9, 7, 7, 5, 4, 8, 8, 9, 7, 5, 1, 3, 3, 8, 7, 8, 3, 6, 5, 0, 8, 3, 5, 3, 3, 8, 6, 2, 7, 3, 4, 7, 5, 1, 0, 8, 3, 4, 5, 1, 0, 3, 0, 7, 0, 5, 5, 4, 6, 4, 3, 4, 1, 2, 9, 7, 0, 8, 3, 4</div> <div class=seqdatalinks> (<a href="/A382688/list">list</a>; <a href="/A382688/constant">constant</a>; <a href="/A382688/graph">graph</a>; <a href="/search?q=A382688+-id:A382688">refs</a>; <a href="/A382688/listen">listen</a>; <a href="/history?seq=A382688">history</a>; <a href="/search?q=id:A382688&fmt=text">text</a>; <a href="/A382688/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>0,1</div> </div> </div> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <div class=sectline>There are floor(k/2) positive zeros of the Legendre polynomial of degree k:</div> <div class=sectline> k | zeros | corresponding weights</div> <div class=sectline>---+---------------------------+--------------------------</div> <div class=sectline> 2 | <a href="/A020760" title="Decimal expansion of 1/sqrt(3).">A020760</a> | <a href="/A000007" title="The characteristic function of {0}: a(n) = 0^n.">A000007</a>*10</div> <div class=sectline> 3 | <a href="/A010513" title="Decimal expansion of square root of 60.">A010513</a>/10 | <a href="/A010716" title="Constant sequence: the all 5's sequence.">A010716</a></div> <div class=sectline> 4 | <a href="/A372267" title="Decimal expansion of the smallest positive zero of the Legendre polynomial of degree 4.">A372267</a>, <a href="/A372268" title="Decimal expansion of the largest positive zero of the Legendre polynomial of degree 4.">A372268</a> | <a href="/A382103" title="Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372267.">A382103</a>, <a href="/A382104" title="Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372268.">A382104</a></div> <div class=sectline> 5 | <a href="/A372269" title="Decimal expansion of the smallest positive zero of the Legendre polynomial of degree 5.">A372269</a>, <a href="/A372270" title="Decimal expansion of the largest positive zero of the Legendre polynomial of degree 5.">A372270</a> | <a href="/A382105" title="Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372269.">A382105</a>, <a href="/A382106" title="Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372270.">A382106</a></div> <div class=sectline> 6 | <a href="/A372271" title="Decimal expansion of the smallest positive zero of the Legendre polynomial of degree 6.">A372271</a>, <a href="/A372272" title="Decimal expansion of the middle positive zero of the Legendre polynomial of degree 6.">A372272</a>, <a href="/A372273" title="Decimal expansion of the largest positive zero of the Legendre polynomial of degree 6.">A372273</a> | <a href="/A382107" title="Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372271.">A382107</a>, <a href="/A382686" title="Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372272.">A382686</a>, <a href="/A382687" title="Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372273.">A382687</a></div> <div class=sectline> 7 | <a href="/A372274" title="Decimal expansion of the smallest positive zero of the Legendre polynomial of degree 7.">A372274</a>, <a href="/A372275" title="Decimal expansion of the middle positive zero of the Legendre polynomial of degree 7.">A372275</a>, <a href="/A372276" title="Decimal expansion of the largest positive zero of the Legendre polynomial of degree 7.">A372276</a> | this sequence, <a href="/A382689" title="Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372275.">A382689</a>, <a href="/A382690" title="Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372276.">A382690</a></div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline><a href="/A382688/b382688.txt">Table of n, a(n) for n=0..82.</a></div> <div class=sectline>A.H.M. Smeets, <a href="/A382103/a382103.py.txt">Python program for Legendre-Gauss quadrature constants.</a></div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Legendre-GaussQuadrature.html">Legendre-Gauss Quadrature</a>.</div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>0.3818300505051189449503697754889751338783650835338627...</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A372274" title="Decimal expansion of the smallest positive zero of the Legendre polynomial of degree 7.">A372274</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="a decimal expansion of a number"><a href="/A382688/constant">cons</a></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:A.H.M._Smeets">A.H.M. Smeets</a>, Apr 03 2025</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="/A382689">A382689</a> </div> <div class=seqname> Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa <a href="/A372275" title="Decimal expansion of the middle positive zero of the Legendre polynomial of degree 7.">A372275</a>. </div> </div> <div class=scorerefs> +0<br> 0 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>2, 7, 9, 7, 0, 5, 3, 9, 1, 4, 8, 9, 2, 7, 6, 6, 6, 7, 9, 0, 1, 4, 6, 7, 7, 7, 1, 4, 2, 3, 7, 7, 9, 5, 8, 2, 4, 8, 6, 9, 2, 5, 0, 6, 5, 2, 2, 6, 5, 9, 8, 7, 6, 4, 5, 3, 7, 0, 1, 4, 0, 3, 2, 6, 9, 3, 6, 1, 8, 8, 1, 0, 4, 3, 0, 5, 6, 2, 6, 7, 6, 8, 1, 3, 2, 4, 0</div> <div class=seqdatalinks> (<a href="/A382689/list">list</a>; <a href="/A382689/constant">constant</a>; <a href="/A382689/graph">graph</a>; <a href="/search?q=A382689+-id:A382689">refs</a>; <a href="/A382689/listen">listen</a>; <a href="/history?seq=A382689">history</a>; <a href="/search?q=id:A382689&fmt=text">text</a>; <a href="/A382689/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>0,1</div> </div> </div> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <div class=sectline>There are floor(k/2) positive zeros of the Legendre polynomial of degree k:</div> <div class=sectline> k | zeros | corresponding weights</div> <div class=sectline>---+---------------------------+--------------------------</div> <div class=sectline> 2 | <a href="/A020760" title="Decimal expansion of 1/sqrt(3).">A020760</a> | <a href="/A000007" title="The characteristic function of {0}: a(n) = 0^n.">A000007</a>*10</div> <div class=sectline> 3 | <a href="/A010513" title="Decimal expansion of square root of 60.">A010513</a>/10 | <a href="/A010716" title="Constant sequence: the all 5's sequence.">A010716</a></div> <div class=sectline> 4 | <a href="/A372267" title="Decimal expansion of the smallest positive zero of the Legendre polynomial of degree 4.">A372267</a>, <a href="/A372268" title="Decimal expansion of the largest positive zero of the Legendre polynomial of degree 4.">A372268</a> | <a href="/A382103" title="Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372267.">A382103</a>, <a href="/A382104" title="Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372268.">A382104</a></div> <div class=sectline> 5 | <a href="/A372269" title="Decimal expansion of the smallest positive zero of the Legendre polynomial of degree 5.">A372269</a>, <a href="/A372270" title="Decimal expansion of the largest positive zero of the Legendre polynomial of degree 5.">A372270</a> | <a href="/A382105" title="Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372269.">A382105</a>, <a href="/A382106" title="Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372270.">A382106</a></div> <div class=sectline> 6 | <a href="/A372271" title="Decimal expansion of the smallest positive zero of the Legendre polynomial of degree 6.">A372271</a>, <a href="/A372272" title="Decimal expansion of the middle positive zero of the Legendre polynomial of degree 6.">A372272</a>, <a href="/A372273" title="Decimal expansion of the largest positive zero of the Legendre polynomial of degree 6.">A372273</a> | <a href="/A382107" title="Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372271.">A382107</a>, <a href="/A382686" title="Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372272.">A382686</a>, <a href="/A382687" title="Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372273.">A382687</a></div> <div class=sectline> 7 | <a href="/A372274" title="Decimal expansion of the smallest positive zero of the Legendre polynomial of degree 7.">A372274</a>, <a href="/A372275" title="Decimal expansion of the middle positive zero of the Legendre polynomial of degree 7.">A372275</a>, <a href="/A372276" title="Decimal expansion of the largest positive zero of the Legendre polynomial of degree 7.">A372276</a> | <a href="/A382688" title="Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372274.">A382688</a>, this sequence, <a href="/A382690" title="Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372276.">A382690</a></div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline><a href="/A382689/b382689.txt">Table of n, a(n) for n=0..86.</a></div> <div class=sectline>A.H.M. Smeets, <a href="/A382103/a382103.py.txt">Python program for Legendre-Gauss quadrature constants.</a></div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Legendre-GaussQuadrature.html">Legendre-Gauss Quadrature</a>.</div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>0.279705391489276667901467771423779582486925065226598764...</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A372275" title="Decimal expansion of the middle positive zero of the Legendre polynomial of degree 7.">A372275</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="a decimal expansion of a number"><a href="/A382689/constant">cons</a></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:A.H.M._Smeets">A.H.M. Smeets</a>, Apr 03 2025</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="/A382690">A382690</a> </div> <div class=seqname> Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa <a href="/A372276" title="Decimal expansion of the largest positive zero of the Legendre polynomial of degree 7.">A372276</a>. </div> </div> <div class=scorerefs> +0<br> 0 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>1, 2, 9, 4, 8, 4, 9, 6, 6, 1, 6, 8, 8, 6, 9, 6, 9, 3, 2, 7, 0, 6, 1, 1, 4, 3, 2, 6, 7, 9, 0, 8, 2, 0, 1, 8, 3, 2, 8, 5, 8, 7, 4, 0, 2, 2, 5, 9, 9, 4, 6, 6, 6, 3, 9, 7, 7, 2, 0, 8, 6, 3, 8, 7, 2, 4, 6, 5, 5, 2, 3, 4, 9, 7, 2, 0, 4, 2, 3, 0, 8, 7, 1, 5, 6, 2, 5</div> <div class=seqdatalinks> (<a href="/A382690/list">list</a>; <a href="/A382690/constant">constant</a>; <a href="/A382690/graph">graph</a>; <a href="/search?q=A382690+-id:A382690">refs</a>; <a href="/A382690/listen">listen</a>; <a href="/history?seq=A382690">history</a>; <a href="/search?q=id:A382690&fmt=text">text</a>; <a href="/A382690/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>COMMENTS</div> <div class=sectbody> <div class=sectline>There are floor(k/2) positive zeros of the Legendre polynomial of degree k:</div> <div class=sectline> k | zeros | corresponding weights</div> <div class=sectline>---+---------------------------+--------------------------</div> <div class=sectline> 2 | <a href="/A020760" title="Decimal expansion of 1/sqrt(3).">A020760</a> | <a href="/A000007" title="The characteristic function of {0}: a(n) = 0^n.">A000007</a>*10</div> <div class=sectline> 3 | <a href="/A010513" title="Decimal expansion of square root of 60.">A010513</a>/10 | <a href="/A010716" title="Constant sequence: the all 5's sequence.">A010716</a></div> <div class=sectline> 4 | <a href="/A372267" title="Decimal expansion of the smallest positive zero of the Legendre polynomial of degree 4.">A372267</a>, <a href="/A372268" title="Decimal expansion of the largest positive zero of the Legendre polynomial of degree 4.">A372268</a> | <a href="/A382103" title="Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372267.">A382103</a>, <a href="/A382104" title="Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372268.">A382104</a></div> <div class=sectline> 5 | <a href="/A372269" title="Decimal expansion of the smallest positive zero of the Legendre polynomial of degree 5.">A372269</a>, <a href="/A372270" title="Decimal expansion of the largest positive zero of the Legendre polynomial of degree 5.">A372270</a> | <a href="/A382105" title="Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372269.">A382105</a>, <a href="/A382106" title="Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372270.">A382106</a></div> <div class=sectline> 6 | <a href="/A372271" title="Decimal expansion of the smallest positive zero of the Legendre polynomial of degree 6.">A372271</a>, <a href="/A372272" title="Decimal expansion of the middle positive zero of the Legendre polynomial of degree 6.">A372272</a>, <a href="/A372273" title="Decimal expansion of the largest positive zero of the Legendre polynomial of degree 6.">A372273</a> | <a href="/A382107" title="Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372271.">A382107</a>, <a href="/A382686" title="Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372272.">A382686</a>, <a href="/A382687" title="Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372273.">A382687</a></div> <div class=sectline> 7 | <a href="/A372274" title="Decimal expansion of the smallest positive zero of the Legendre polynomial of degree 7.">A372274</a>, <a href="/A372275" title="Decimal expansion of the middle positive zero of the Legendre polynomial of degree 7.">A372275</a>, <a href="/A372276" title="Decimal expansion of the largest positive zero of the Legendre polynomial of degree 7.">A372276</a> | <a href="/A382688" title="Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372274.">A382688</a>, <a href="/A382689" title="Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372275.">A382689</a>, this sequence</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline><a href="/A382690/b382690.txt">Table of n, a(n) for n=0..86.</a></div> <div class=sectline>A.H.M. Smeets, <a href="/A382103/a382103.py.txt">Python program for Legendre-Gauss quadrature constants.</a></div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Legendre-GaussQuadrature.html">Legendre-Gauss Quadrature</a>.</div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>0.12948496616886969327061143267908201832858740225994666...</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A372276" title="Decimal expansion of the largest positive zero of the Legendre polynomial of degree 7.">A372276</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="a decimal expansion of a number"><a href="/A382690/constant">cons</a></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:A.H.M._Smeets">A.H.M. Smeets</a>, Apr 03 2025</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="/A382364">A382364</a> </div> <div class=seqname> a(n) is the smallest squarefree number k such that the sum of the digit counts of the prime factors of k equals the sum of n and the digit count of k </div> </div> <div class=scorerefs> +0<br> 0 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>6, 66, 858, 72930, 6374082, 643782282, 66309575046</div> <div class=seqdatalinks> (<a href="/A382364/list">list</a>; <a href="/A382364/graph">graph</a>; <a href="/search?q=A382364+-id:A382364">refs</a>; <a href="/A382364/listen">listen</a>; <a href="/history?seq=A382364">history</a>; <a href="/search?q=id:A382364&fmt=text">text</a>; <a href="/A382364/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="/A382364/b382364.txt">Table of n, a(n) for n=1..7.</a></div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>a(1) = 6 = 2*3, because the total number of digits in its distinct prime factors (2 and 3) is 2. This equals the sum of n = 1 and the number of digits in 6, which is 1, and no lesser number has this property.</div> <div class=sectline>a(2) = 66 = 2*3*11, because the total number of digits in its distinct prime factors (2, 3 and 11) is 4. This equals the sum of n = 2 and the number of digits in 66, which is 2, and no lesser number has this property.</div> <div class=sectline>Table begins:</div> <div class=sectline> 1 6 = 2 * 3;</div> <div class=sectline> 2 66 = 2 * 3 * 11;</div> <div class=sectline> 3 858 = 2 * 3 * 11 * 13;</div> <div class=sectline> 4 72930 = 2 * 3 * 5 * 11 * 13 * 17;</div> <div class=sectline> 5 6374082 = 2 * 3 * 11 * 13 * 17 * 19 * 23;</div> <div class=sectline> 6 643782282 = 2 * 3 * 11 * 13 * 17 * 19 * 23 * 101;</div> <div class=sectline> 7 66309575046 = 2 * 3 * 11 * 13 * 17 * 19 * 23 * 101 * 103;</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(PARI) isok(k, n) = if (issquarefree(k), my(f=factor(k)[, 1]); sum(i=1, #f, #digits(f[i])) == n+#digits(k));</div> <div class=sectline>a(n) = my(k=2); while (!isok(k, n), k++); k; \\ <a href="/wiki/User:Michel_Marcus">Michel Marcus</a>, Apr 02 2025</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A055642" title="Number of digits in the decimal expansion of n.">A055642</a>, <a href="/A095411" title="Numbers k such that total number of decimal digits of all distinct prime factors of k is larger than the number of digits of k.">A095411</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="dependent on base used for sequence">base</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:Jean-Marc_Rebert">Jean-Marc Rebert</a>, Mar 24 2025</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="/A382185">A382185</a> </div> <div class=seqname> a(n) is the n-th tribonacci number modulo the n-th prime. </div> </div> <div class=scorerefs> +0<br> 0 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>0, 1, 1, 2, 4, 7, 13, 5, 21, 23, 25, 15, 12, 24, 13, 9, 45, 56, 16, 35, 20, 71, 47, 9, 40, 80, 18, 46, 75, 101, 55, 48, 65, 36, 142, 34, 91, 0, 43, 147, 118, 41, 175, 24, 131, 152, 189, 213, 116, 201, 116, 66, 73, 9, 0, 53, 210, 239, 167, 171, 87, 262, 251, 111, 115, 69, 284, 186, 211, 321, 331, 135</div> <div class=seqdatalinks> (<a href="/A382185/list">list</a>; <a href="/A382185/graph">graph</a>; <a href="/search?q=A382185+-id:A382185">refs</a>; <a href="/A382185/listen">listen</a>; <a href="/history?seq=A382185">history</a>; <a href="/search?q=id:A382185&fmt=text">text</a>; <a href="/A382185/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>For n=1..10^6, a(n) is 0 only for n=1, n=38, n=55, n=203, n=51658.</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline><a href="/A382185/b382185.txt">Table of n, a(n) for n=1..72.</a></div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>a(n) = <a href="/A000073" title="Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) for n >= 3 with a(0) = a(1) = 0 and a(2) = 1.">A000073</a>(n) mod <a href="/A000040" title="The prime numbers.">A000040</a>(n).</div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>a(10) = <a href="/A000073" title="Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) for n >= 3 with a(0) = a(1) = 0 and a(2) = 1.">A000073</a>(10) mod prime(10) = 81 mod 29 = 23.</div> </div> </div> <div class=section> <div class=sectname>MAPLE</div> <div class=sectbody> <div class=sectline>a:= n-> (<<0|1|0>, <0|0|1>, <1|1|1>>^n mod ithprime(n))[1, 3]:</div> <div class=sectline>seq(a(n), n=1..100); # <a href="/wiki/User:Alois_P._Heinz">Alois P. Heinz</a>, Apr 03 2025</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(PARI) a(n) = ([0, 1, 0; 0, 0, 1; 1, 1, 1]^n)[1, 3] % prime(n); \\ <a href="/wiki/User:Michel_Marcus">Michel Marcus</a>, Apr 02 2025</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A000040" title="The prime numbers.">A000040</a>, <a href="/A000073" title="Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) for n >= 3 with a(0) = a(1) = 0 and a(2) = 1.">A000073</a>, <a href="/A072123" title="Remainder when Fibonacci(n) is divided by prime(n).">A072123</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:Michael_Figelius">Michael Figelius</a>, Mar 17 2025</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="/A382623">A382623</a> </div> <div class=seqname> Positive integers that contain an even digit d immediately preceded by a digit <= d. </div> </div> <div class=scorerefs> +0<br> 0 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>12, 14, 16, 18, 22, 24, 26, 28, 34, 36, 38, 44, 46, 48, 56, 58, 66, 68, 78, 88, 100, 102, 104, 106, 108, 112, 114, 116, 118, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 134, 136, 138, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 156, 158, 160, 161, 162, 163</div> <div class=seqdatalinks> (<a href="/A382623/list">list</a>; <a href="/A382623/graph">graph</a>; <a href="/search?q=A382623+-id:A382623">refs</a>; <a href="/A382623/listen">listen</a>; <a href="/history?seq=A382623">history</a>; <a href="/search?q=id:A382623&fmt=text">text</a>; <a href="/A382623/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>Conjecture: these are the numbers missing from <a href="/A382621" title="Lexicographically earliest sequence of distinct positive integers such that if a digit d in the digit stream (ignoring comma...">A382621</a>.</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>Paolo Xausa, <a href="/A382623/b382623.txt">Table of n, a(n) for n = 1..10000</a></div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>A382623Q[k_] := MemberQ[Partition[IntegerDigits[k], 2, 1], {i_, j_?EvenQ} /; i <= j];</div> <div class=sectline>Select[Range[200], A382623Q]</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(Python)</div> <div class=sectline>def ok(n):</div> <div class=sectline> s = str(n)</div> <div class=sectline> return any(s[i+1] >= s[i] for i in range(len(s)-1) if s[i+1] in "02468")</div> <div class=sectline>print([k for k in range(1, 164) if ok(k)]) # <a href="/wiki/User:Michael_S._Branicky">Michael S. Branicky</a>, Apr 03 2025</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A347298" title="Numbers that contain an even digit d immediately followed by a digit <= d.">A347298</a>, <a href="/A382464" title="Positive integers that contain an even digit d immediately preceded by a digit >= d.">A382464</a>, <a href="/A382621" title="Lexicographically earliest sequence of distinct positive integers such that if a digit d in the digit stream (ignoring comma...">A382621</a>, <a href="/A382624" title="Positive integers such that every even digit except the leftmost is immediately preceded by a larger digit.">A382624</a> (complement).</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="dependent on base used for sequence">base</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:Paolo_Xausa">Paolo Xausa</a>, Apr 01 2025</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="/A382622">A382622</a> </div> <div class=seqname> First differences of <a href="/A382621" title="Lexicographically earliest sequence of distinct positive integers such that if a digit d in the digit stream (ignoring comma...">A382621</a>. </div> </div> <div class=scorerefs> +0<br> 0 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>2, -1, 3, -1, 3, -1, 3, -1, 2, 1, 2, 2, 2, 2, 1, 10, 1, 1, 1, -12, 14, -12, 2, 2, 2, 8, 2, 1, 10, 1, 1, 1, 1, 1, -14, 16, -15, 17, -16, 27, 1, 1, 1, 1, 1, -30, 2, 2, 11, 16, 1, -16, 18, -17, 28, 1, 1, 1, 1, 1, 1, 1, -34, 35, -34, 35, -34, 36, 2, 2, 2, -40, 2, 11, 29</div> <div class=seqdatalinks> (<a href="/A382622/list">list</a>; <a href="/A382622/graph">graph</a>; <a href="/search?q=A382622+-id:A382622">refs</a>; <a href="/A382622/listen">listen</a>; <a href="/history?seq=A382622">history</a>; <a href="/search?q=id:A382622&fmt=text">text</a>; <a href="/A382622/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>Paolo Xausa, <a href="/A382622/b382622.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="/A382621" title="Lexicographically earliest sequence of distinct positive integers such that if a digit d in the digit stream (ignoring comma...">A382621</a>(n+1) - <a href="/A382621" title="Lexicographically earliest sequence of distinct positive integers such that if a digit d in the digit stream (ignoring comma...">A382621</a>(n).</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>(* A382621list is defined at <a href="/A382621" title="Lexicographically earliest sequence of distinct positive integers such that if a digit d in the digit stream (ignoring comma...">A382621</a> *)</div> <div class=sectline>Differences[A382621list[100]]</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A382621" title="Lexicographically earliest sequence of distinct positive integers such that if a digit d in the digit stream (ignoring comma...">A382621</a>.</div> </div> </div> <div class=section> <div class=sectname>KEYWORD</div> <div class=sectbody> <div class=sectline><span title="sequence contains negative numbers">sign</span>,<span title="dependent on base used for sequence">base</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:Paolo_Xausa">Paolo Xausa</a>, Apr 01 2025</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=500&sort=created">51</a> </font> </div> </div> </div> <div class=resultline></div> </div> <center><p><font size=-1> Search completed in 0.153 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 April 9 02:56 EDT 2025. 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