A344687 - 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>A344687 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A344687" 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=%2fA344687">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="A344687 - 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> A344687 </div> <div class=seqname> a(n) is the lowest nonnegative exponent k such that n!^k is the product of the divisors of n!. </div> </div> <div class=scorerefs> 1 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>0, 1, 2, 4, 8, 15, 30, 48, 80, 135, 270, 396, 792, 1296, 2016, 2688, 5376, 7344, 14688, 20520, 30400, 48000, 96000, 121440, 170016, 266112, 338688, 458640, 917280, 1166400, 2332800, 2764800, 3932160, 6082560, 8211456, 9797760, 19595520, 30233088, 42550272</div> <div class=seqdatalinks> (<a href="/A344687/list">list</a>; <a href="/A344687/graph">graph</a>; <a href="/search?q=A344687+-id:A344687">refs</a>; <a href="/A344687/listen">listen</a>; <a href="/history?seq=A344687">history</a>; <a href="/search?q=id:A344687&fmt=text">text</a>; <a href="/A344687/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>1,3</div> </div> </div> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <div class=sectline>This sequence is a subsequence of <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>, because the product of divisors of n! is n^(d(n)/2) (where d(n) is the number of divisors of n), so a(n) = d(n!)/2.</div> <div class=sectline>For prime p, d(p!) = 2*d((p-1)!), so a(p) = 2*a(p-1).</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline><a href="/A344687/b344687.txt">Table of n, a(n) for n=1..39.</a></div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>a(n) = d(n!)/2 = <a href="/A000005" title="d(n) (also called tau(n) or sigma_0(n)), the number of divisors of n.">A000005</a>(<a href="/A000142" title="Factorial numbers: n! = 1*2*3*4*...*n (order of symmetric group S_n, number of permutations of n letters).">A000142</a>(n))/2 = <a href="/A027423" title="Number of divisors of n!.">A027423</a>(n)/2 for n > 1.</div> <div class=sectline>a(n) = <a href="/A157672" title="Number of unordered factorizations of n! into two distinct proper factors.">A157672</a>(n-1) + 1 for all n >= 2.</div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>For n = 4, n! = 24 = 2^3 * 3, which has (3+1)*(1+1) = 8 divisors: {1,2,3,4,6,8,12,24} whose product is 331776 = (24)^4 = (4!)^4. So a(4) = 4.</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>Join[{0}, Table[DivisorSigma[0, n!]/2, {n, 2, 39}]] (* <a href="/wiki/User:Stefano_Spezia">Stefano Spezia</a>, Aug 18 2021 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(Python)</div> <div class=sectline>def a(n):</div> <div class=sectline> d = {}</div> <div class=sectline> for i in range(2, n+1):</div> <div class=sectline> tmp = i</div> <div class=sectline> j = 2</div> <div class=sectline> while(tmp != 1):</div> <div class=sectline> if(tmp % j == 0):</div> <div class=sectline> d.setdefault(j, 0)</div> <div class=sectline> tmp //= j</div> <div class=sectline> d[j] += 1</div> <div class=sectline> else:</div> <div class=sectline> j += 1</div> <div class=sectline> res = 1</div> <div class=sectline> for i in d.values():</div> <div class=sectline> res *= (i+1)</div> <div class=sectline> return res // 2</div> <div class=sectline>(Python)</div> <div class=sectline>from math import prod</div> <div class=sectline>from collections import Counter</div> <div class=sectline>from sympy import factorint</div> <div class=sectline>def <a href="/A344687" title="a(n) is the lowest nonnegative exponent k such that n!^k is the product of the divisors of n!.">A344687</a>(n): return prod(e+1 for e in sum((Counter(factorint(i)) for i in range(2, n+1)), start=Counter()).values())//2 # <a href="/wiki/User:Chai_Wah_Wu">Chai Wah Wu</a>, Jun 25 2022</div> <div class=sectline>(PARI) a(n) = if (n==1, 0, numdiv(n!)/2); \\ <a href="/wiki/User:Michel_Marcus">Michel Marcus</a>, Aug 18 2021</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A000005" title="d(n) (also called tau(n) or sigma_0(n)), the number of divisors of n.">A000005</a>, <a href="/A000142" title="Factorial numbers: n! = 1*2*3*4*...*n (order of symmetric group S_n, number of permutations of n letters).">A000142</a>, <a href="/A027423" title="Number of divisors of n!.">A027423</a>, <a href="/A280420" title="Product of divisors of n!.">A280420</a>, <a href="/A157672" title="Number of unordered factorizations of n! into two distinct proper factors.">A157672</a>.</div> <div class=sectline>Sequence in context: <a href="/A176503" title="Leading column of triangle in A176463.">A176503</a> <a href="/A262333" title="Number of (n+3) X (1+3) 0..1 arrays with each row and column divisible by 9, read as a binary number with top and left being...">A262333</a> <a href="/A293335" title="Least integer k such that k/2^n > sqrt(1/5).">A293335</a> * <a href="/A301480" title="Number of rooted twice-partitions of n.">A301480</a> <a href="/A217777" title="Expansion of (1+x)*(1+2*x)*(1-x)/(1-5*x^2+5*x^4).">A217777</a> <a href="/A034338" title="Number of binary [ n,4 ] codes of dimension <= 4 without zero columns.">A034338</a></div> <div class=sectline>Adjacent sequences: <a href="/A344684" title="Sum of two consecutive products of Fibonacci and Pell numbers: F(n)*P(n) + F(n+1)*P(n+1).">A344684</a> <a href="/A344685" title="Triangle T(n, k) obtained from the array N1(a, b) = a^2 + a*b - b^2, for a >= 0 and b >= 0, read by upwards antidiagonals.">A344685</a> <a href="/A344686" title="Triangle T(n, k) obtained from the array N2(a, b) = a^2 - a*b - b^2, for a >= 0 and b >= 0, read by upwards antidiagonals.">A344686</a> * <a href="/A344688" title="Decimal expansion of 3236 * Pi^9 / (55801305 * sqrt(3)).">A344688</a> <a href="/A344689" title="a(n) is the number of preference profiles in the stable marriage problem with n men and n women such that one man and one wo...">A344689</a> <a href="/A344690" title="a(n) is the number of multisets of size n consisting of permutations of n elements.">A344690</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:Alex_Sokolov">Alex Sokolov</a>, Aug 17 2021</div> </div> </div> <div class=section> <div class=sectname>STATUS</div> <div class=sectbody> <div class=sectline>approved</div> </div> </div> </div> <div class=space10></div> </div> </div></div> <p> <div class=footerpad></div> <div class=footer> <center> <div class=bottom> <div class=linksbar> <a href="/">Lookup</a> <a href="/wiki/Welcome"><font color="red">Welcome</font></a> <a href="/wiki/Main_Page"><font color="red">Wiki</font></a> <a href="/wiki/Special:RequestAccount">Register</a> <a href="/play.html">Music</a> <a href="/plot2.html">Plot 2</a> <a href="/demo1.html">Demos</a> <a 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 28 04:31 EST 2024. Contains 378181 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>

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A344687 - OEIS