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A300836 - OEIS
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 3.2 Final//EN"> <html> <head> <link rel="stylesheet" href="/styles.css"> <meta name="format-detection" content="telephone=no"> <meta http-equiv="content-type" content="text/html; charset=utf-8"> <meta name=viewport content="width=device-width, initial-scale=1"> <meta name="keywords" content="OEIS,integer sequences,Sloane" /> <title>A300836 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A300836" 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=%2fA300836">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="A300836 - OEIS"></a> </div> </center> </div></div> <div class=center><div class=pagebody> <div class=searchbarcenter> <form name=f action="/search" method="GET"> <div class=searchbargreet> <div class=searchbar> <div class=searchq> <input class=searchbox maxLength=1024 name=q value="" title="Search Query"> </div> <div class=searchsubmit> <input type=submit value="Search" name=go> </div> <div class=hints> <span class=hints><a href="/hints.html">Hints</a></span> </div> </div> <div class=searchgreet> (Greetings from <a href="/welcome">The On-Line Encyclopedia of Integer Sequences</a>!) </div> </div> </form> </div> <div class=sequence> <div class=space1></div> <div class=line></div> <div class=seqhead> <div class=seqnumname> <div class=seqnum> A300836 </div> <div class=seqname> a(n) is the total number of terms (1-digits) in Zeckendorf representation of all proper divisors of n. </div> </div> <div class=scorerefs> 8 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>0, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 7, 1, 4, 3, 5, 1, 7, 1, 7, 4, 4, 1, 11, 2, 3, 4, 8, 1, 10, 1, 7, 4, 5, 4, 14, 1, 5, 3, 11, 1, 10, 1, 8, 7, 4, 1, 15, 3, 8, 5, 7, 1, 12, 4, 12, 5, 4, 1, 21, 1, 5, 7, 10, 3, 13, 1, 8, 4, 11, 1, 19, 1, 4, 8, 10, 5, 10, 1, 16, 7, 5, 1, 20, 5, 5, 4, 12, 1, 20, 4, 10, 5, 4, 5, 21, 1, 9, 10, 16, 1, 13, 1, 11, 10</div> <div class=seqdatalinks> (<a href="/A300836/list">list</a>; <a href="/A300836/graph">graph</a>; <a href="/search?q=A300836+-id:A300836">refs</a>; <a href="/A300836/listen">listen</a>; <a href="/history?seq=A300836">history</a>; <a href="/search?q=id:A300836&fmt=text">text</a>; <a href="/A300836/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>LINKS</div> <div class=sectbody> <div class=sectline>Antti Karttunen, <a href="/A300836/b300836.txt">Table of n, a(n) for n = 1..65537</a></div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>a(n) = Sum_{d|n, d<n} <a href="/A007895" title="Number of terms in the Zeckendorf representation of n (write n as a sum of non-consecutive distinct Fibonacci numbers).">A007895</a>(d).</div> <div class=sectline>a(n) = <a href="/A300837" title="a(n) is the total number of terms (1-digits) in Zeckendorf representation of all divisors of n.">A300837</a>(n) - <a href="/A007895" title="Number of terms in the Zeckendorf representation of n (write n as a sum of non-consecutive distinct Fibonacci numbers).">A007895</a>(n).</div> <div class=sectline>a(n) = <a href="/A001222" title="Number of prime divisors of n counted with multiplicity (also called big omega of n, bigomega(n) or Omega(n)).">A001222</a>(<a href="/A300834" title="a(n) = Product_{d|n, d<n} A019565(A003714(d)), where A003714(n) is the n-th Fibbinary number.">A300834</a>(n)).</div> <div class=sectline>For all n >=1, a(n) >= <a href="/A293435" title="a(n) is the number of the proper divisors of n that are Fibonacci numbers (A000045).">A293435</a>(n).</div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>For n=12, its proper divisors are 1, 2, 3, 4 and 6. Zeckendorf-representations (<a href="/A014417" title="Representation of n in base of Fibonacci numbers (the Zeckendorf representation of n). Also, binary words starting with 1 no...">A014417</a>) of these numbers are 1, 10, 100, 101 and 1001. Total number of 1's present is 7, thus a(12) = 7.</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(PARI)</div> <div class=sectline><a href="/A072649" title="n occurs Fibonacci(n) times (cf. A000045).">A072649</a>(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From <a href="/A072649" title="n occurs Fibonacci(n) times (cf. A000045).">A072649</a></div> <div class=sectline><a href="/A007895" title="Number of terms in the Zeckendorf representation of n (write n as a sum of non-consecutive distinct Fibonacci numbers).">A007895</a>(n) = { my(s=0); while(n>0, s++; n -= fibonacci(1+<a href="/A072649" title="n occurs Fibonacci(n) times (cf. A000045).">A072649</a>(n))); (s); }</div> <div class=sectline><a href="/A300836" title="a(n) is the total number of terms (1-digits) in Zeckendorf representation of all proper divisors of n.">A300836</a>(n) = sumdiv(n, d, (d<n)*<a href="/A007895" title="Number of terms in the Zeckendorf representation of n (write n as a sum of non-consecutive distinct Fibonacci numbers).">A007895</a>(d));</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <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="/A007895" title="Number of terms in the Zeckendorf representation of n (write n as a sum of non-consecutive distinct Fibonacci numbers).">A007895</a>, <a href="/A014417" title="Representation of n in base of Fibonacci numbers (the Zeckendorf representation of n). Also, binary words starting with 1 no...">A014417</a>, <a href="/A072649" title="n occurs Fibonacci(n) times (cf. A000045).">A072649</a>, <a href="/A300834" title="a(n) = Product_{d|n, d<n} A019565(A003714(d)), where A003714(n) is the n-th Fibbinary number.">A300834</a>, <a href="/A300837" title="a(n) is the total number of terms (1-digits) in Zeckendorf representation of all divisors of n.">A300837</a>.</div> <div class=sectline>Cf. also <a href="/A292257" title="a(n) is the total number of 1's in binary expansion of all proper divisors of n.">A292257</a>, <a href="/A293435" title="a(n) is the number of the proper divisors of n that are Fibonacci numbers (A000045).">A293435</a>.</div> <div class=sectline>Sequence in context: <a href="/A334033" title="The a(n)-th composition in standard order (graded reverse-lexicographic) is the reversed unsorted prime signature of n.">A334033</a> <a href="/A339564" title="Number of ways to choose a distinct factor in a factorization of n (pointed factorizations).">A339564</a> <a href="/A296119" title="Number of ways to choose a strict factorization of each factor in a factorization of n.">A296119</a> * <a href="/A118314" title="Erroneous version of A002033.">A118314</a> <a href="/A349059" title="Number of weakly alternating ordered factorizations of n.">A349059</a> <a href="/A002033" title="Number of perfect partitions of n.">A002033</a></div> <div class=sectline>Adjacent sequences: <a href="/A300833" title="Filter sequence combining A300830(n), A300831(n) and A300832(n), three products formed from such proper divisors d of n for ...">A300833</a> <a href="/A300834" title="a(n) = Product_{d|n, d<n} A019565(A003714(d)), where A003714(n) is the n-th Fibbinary number.">A300834</a> <a href="/A300835" title="Restricted growth sequence transform of A300834, product_{d|n, d<n} A019565(A003714(d)); Filter sequence related to Zeckendo...">A300835</a> * <a href="/A300837" title="a(n) is the total number of terms (1-digits) in Zeckendorf representation of all divisors of n.">A300837</a> <a href="/A300838" title="Permutation of nonnegative integers: a(n) = A057300(A003188(n)).">A300838</a> <a href="/A300839" title="Permutation of nonnegative integers: a(n) = A006068(A057300(n)).">A300839</a></div> </div> </div> <div class=section> <div class=sectname>KEYWORD</div> <div class=sectbody> <div class=sectline><span title="a sequence of nonnegative numbers">nonn</span></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:Antti_Karttunen">Antti Karttunen</a>, Mar 18 2018</div> </div> </div> <div class=section> <div class=sectname>STATUS</div> <div class=sectbody> <div class=sectline>approved</div> </div> </div> </div> <div class=space10></div> </div> </div></div> <p> <div class=footerpad></div> <div class=footer> <center> <div class=bottom> <div class=linksbar> <a href="/">Lookup</a> <a href="/wiki/Welcome"><font color="red">Welcome</font></a> <a href="/wiki/Main_Page"><font color="red">Wiki</font></a> <a href="/wiki/Special:RequestAccount">Register</a> <a href="/play.html">Music</a> <a href="/plot2.html">Plot 2</a> <a href="/demo1.html">Demos</a> <a href="/wiki/Index_to_OEIS">Index</a> <a href="/webcam">WebCam</a> <a href="/Submit.html">Contribute</a> <a href="/eishelp2.html">Format</a> <a href="/wiki/Style_Sheet">Style Sheet</a> <a href="/transforms.html">Transforms</a> <a href="/ol.html">Superseeker</a> <a href="/recent">Recents</a> </div> <div class=linksbar> <a href="/community.html">The OEIS Community</a> </div> <div class=linksbar> Maintained by <a href="http://oeisf.org">The OEIS Foundation Inc.</a> </div> <div class=dbinfo>Last modified April 10 10:17 EDT 2025. Contains 382671 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>