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A378168 - 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>A378168 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A378168" 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=%2fA378168">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="A378168 - 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> A378168 </div> <div class=seqname> a(n) is the number of squares <= 10^n that are not higher powers, i.e., terms of <a href="/A076467" title="Perfect powers m^k where m is a positive integer and k >= 3.">A076467</a>. </div> </div> <div class=scorerefs> 3 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>2, 6, 24, 87, 292, 959, 3089, 9875, 31410, 99633, 315589, 998889, 3160340, 9996605, 31616816, 99989509, 316209268, 999967330, 3162219896, 9999897769, 31622595517, 99999679010, 316227196708, 999998989804, 3162275866962, 9999996815862, 31622770946248, 99999989953079</div> <div class=seqdatalinks> (<a href="/A378168/list">list</a>; <a href="/A378168/graph">graph</a>; <a href="/search?q=A378168+-id:A378168">refs</a>; <a href="/A378168/listen">listen</a>; <a href="/history?seq=A378168">history</a>; <a href="/search?q=id:A378168&fmt=text">text</a>; <a href="/A378168/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>Chai Wah Wu, <a href="/A378168/b378168.txt">Table of n, a(n) for n = 1..2000</a></div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>a(n) = Sum_{k=1..floor(log_2(10^n)-1)} mu(k)*floor(10^(n/(2k))-1). - <a href="/wiki/User:Chai_Wah_Wu">Chai Wah Wu</a>, Nov 20 2024</div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>a(1) = 2: squares <= 10 are 2^2 and 3^2;</div> <div class=sectline>a(2) = 6: 2 squares <= 10 and 5^2, 6^2, 7^2, 10^2, but not 4^2=2^4, 8^2=2^6, and 9^2=3^4;</div> <div class=sectline>a(3) = 24: 6 squares <= 100 and all squares between 11^2 and 31^2, except for 16^2=2^8, 25^2=5^4, and 27^2=3^6.</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>Table[Sum[MoebiusMu[k]*Floor[10^(n/(2k))-1], {k, Floor[Log2[10^n]-1]}], {n, 28}] (* <a href="/wiki/User:James_C._McMahon">James C. McMahon</a>, Nov 21 2024 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(Python)</div> <div class=sectline>from math import gcd</div> <div class=sectline>from sympy import integer_nthroot, mobius</div> <div class=sectline>def <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>(n): return sum(mobius(k)*(integer_nthroot(10**(n//(a:=gcd(n, b:=k<<1))), b//a)[0]-1) for k in range(1, (10**n).bit_length()-1)) # <a href="/wiki/User:Chai_Wah_Wu">Chai Wah Wu</a>, Nov 20 2024</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A000290" title="The squares: a(n) = n^2.">A000290</a>, <a href="/A001597" title="Perfect powers: m^k where m > 0 and k >= 2.">A001597</a>, <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="/A089579" title="Total number of perfect powers > 1 below 10^n.">A089579</a>.</div> <div class=sectline>Sequence in context: <a href="/A048120" title="a(n)=T(2n,n+1), where T is given by A048113.">A048120</a> <a href="/A322198" title="a(n) is the coefficient of x^n*y^n in Product_{n>=1} 1/(1 - x^(2*n-1) - y^(2*n-1)).">A322198</a> <a href="/A003759" title="Number of Hamiltonian cycles in D_4 X P_n.">A003759</a> * <a href="/A217527" title="a(n) = 2^(n-2)*(n-2)^2+2^(n-1).">A217527</a> <a href="/A293774" title="Number of permutations of length n sortable by 3 passes through a pop-stack.">A293774</a> <a href="/A226037" title="a(n) = Sum_{c in P(n)} lcm(c) where P(n) is the set of all subsets of {1,2,...,n}.">A226037</a></div> <div class=sectline>Adjacent sequences: <a href="/A378165" title="Differences between adjacent terms of A076467 that correspond to the locations of abc-quality records of A378164.">A378165</a> <a href="/A378166" title="Terms c = A076467(k) such that the distinct prime factors of b = A076467(k-1) and of c-b are subsets of the prime factors of...">A378166</a> <a href="/A378167" title="Differences between adjacent terms of A076467 that correspond to the locations described by A378166.">A378167</a> * <a href="/A378169" title="Number of free polyominoes with n cells with at most 3 collinear cell centers on any line in the plane.">A378169</a> <a href="/A378170" title="Number of subsets of the first n nonzero tetrahedral numbers whose sum is a nonzero tetrahedral number.">A378170</a> <a href="/A378171" title="Number of subsets of the first n positive cubes whose sum is a positive cube.">A378171</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:Hugo_Pfoertner">Hugo Pfoertner</a>, Nov 20 2024</div> </div> </div> <div class=section> <div class=sectname>EXTENSIONS</div> <div class=sectbody> <div class=sectline>a(20) onwards from <a href="/wiki/User:Chai_Wah_Wu">Chai Wah Wu</a>, Nov 20 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></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 February 17 17:38 EST 2025. Contains 380975 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>