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A049363 - 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>A049363 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A049363" 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=%2fA049363">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="A049363 - 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> A049363 </div> <div class=seqname> a(1) = 1; for n > 1, smallest digitally balanced number in base n. </div> </div> <div class=scorerefs> 24 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>1, 2, 11, 75, 694, 8345, 123717, 2177399, 44317196, 1023456789, 26432593615, 754777787027, 23609224079778, 802772380556705, 29480883458974409, 1162849439785405935, 49030176097150555672, 2200618769387072998445, 104753196945250864004691, 5271200265927977839335179</div> <div class=seqdatalinks> (<a href="/A049363/list">list</a>; <a href="/A049363/graph">graph</a>; <a href="/search?q=A049363+-id:A049363">refs</a>; <a href="/A049363/listen">listen</a>; <a href="/history?seq=A049363">history</a>; <a href="/search?q=id:A049363&fmt=text">text</a>; <a href="/A049363/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><a href="/A037968" title="Greatest number of distinct digits of n in any base b>=2.">A037968</a>(a(n)) = n and <a href="/A037968" title="Greatest number of distinct digits of n in any base b>=2.">A037968</a>(m) < n for m < a(n). - <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Oct 27 2003</div> <div class=sectline>Also smallest pandigital number in base n. - <a href="/wiki/User:Franklin_T._Adams-Watters">Franklin T. Adams-Watters</a>, Nov 15 2006</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>Reinhard Zumkeller, <a href="/A049363/b049363.txt">Table of n, a(n) for n = 1..250</a></div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PandigitalNumber.html">Pandigital Number</a></div> <div class=sectline>Wikipedia, <a href="http://en.wikipedia.org/wiki/Pandigital_number">Pandigital number</a></div> <div class=sectline>Chai Wah Wu, <a href="https://arxiv.org/abs/2403.20304">Pandigital and penholodigital numbers</a>, arXiv:2403.20304 [math.GM], 2024. See p. 1.</div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>a(n) = (102345....n-1) in base n. - Ulrich Schimke (ulrschimke(AT)aol.com)</div> <div class=sectline>For n > 1, a(n) = (n^n-n)/(n-1)^2 + n^(n-2)*(n-1) - 1 = <a href="/A023811" title="Largest metadrome (number with digits in strict ascending order) in base n.">A023811</a>(n) + <a href="/A053506" title="a(n) = (n-1)*n^(n-2).">A053506</a>(n). - <a href="/wiki/User:Franklin_T._Adams-Watters">Franklin T. Adams-Watters</a>, Nov 15 2006</div> <div class=sectline>a(n) = n^(n-1) + Sum_{m=2..n-1} m * n^(n - 1 - m). - <a href="/wiki/User:Alexander_R._Povolotsky">Alexander R. Povolotsky</a>, Sep 18 2022</div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>a(6) = 102345_6 = 1*6^5 + 2*6^3 + 3*6^2 + 4*6^1 + 5*6^0 = 8345.</div> </div> </div> <div class=section> <div class=sectname>MAPLE</div> <div class=sectbody> <div class=sectline>a:= n-> n^(n-1)+add((n-i)*n^(i-1), i=1..n-2):</div> <div class=sectline>seq(a(n), n=1..23); # <a href="/wiki/User:Alois_P._Heinz">Alois P. Heinz</a>, May 02 2020</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>Table[FromDigits[Join[{1, 0}, Range[2, n-1]], n], {n, 20}] (* <a href="/wiki/User:Harvey_P._Dale">Harvey P. Dale</a>, Oct 12 2012 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(PARI) <a href="/A049363" title="a(1) = 1; for n > 1, smallest digitally balanced number in base n.">A049363</a>(n)=n^(n-1)+sum(i=1, n-2, n^(i-1)*(n-i)) \\ <a href="/wiki/User:M._F._Hasler">M. F. Hasler</a>, Jan 10 2012</div> <div class=sectline>(PARI) <a href="/A049363" title="a(1) = 1; for n > 1, smallest digitally balanced number in base n.">A049363</a>(n)=if(n>1, (n^n-n)/(n-1)^2+n^(n-2)*(n-1)-1, 1) \\ <a href="/wiki/User:M._F._Hasler">M. F. Hasler</a>, Jan 12 2012</div> <div class=sectline>(Haskell)</div> <div class=sectline>a049363 n = foldl (\v d -> n * v + d) 0 (1 : 0 : [2..n-1])</div> <div class=sectline>-- <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Apr 04 2012</div> <div class=sectline>(Python)</div> <div class=sectline>def <a href="/A049363" title="a(1) = 1; for n > 1, smallest digitally balanced number in base n.">A049363</a>(n): return (n**n-n)//(n-1)**2+n**(n-2)*(n-1)-1 if n>1 else 1 # <a href="/wiki/User:Chai_Wah_Wu">Chai Wah Wu</a>, Mar 13 2024</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Column k=1 of <a href="/A061845" title="Numbers that have one of every digit in some base.">A061845</a> and <a href="/A378000" title="Array read by ascending antidiagonals: T(n,k) is the k-th positive integer that is digitally balanced in base n.">A378000</a> (for n>1).</div> <div class=sectline>Cf. <a href="/A031443" title="Digitally balanced numbers: positive numbers that in base 2 have the same number of 0's as 1's.">A031443</a>, <a href="/A049354" title="Digitally balanced numbers in base 3: equal numbers of 0's, 1's, 2's.">A049354</a>, <a href="/A049355" title="Digitally balanced numbers in base 4: equal numbers of 0's, 1's, ... 3's.">A049355</a>, <a href="/A023811" title="Largest metadrome (number with digits in strict ascending order) in base n.">A023811</a>.</div> <div class=sectline>Sequence in context: <a href="/A198088" title="Number of nXnXn 0..6 triangular arrays with each element x equal to the number its neighbors equal to 4,4,2,0,1,2,0 for x=0,...">A198088</a> <a href="/A112894" title="a(n)*n = A112893(n).">A112894</a> <a href="/A220878" title="Number of Catalan M-paths.">A220878</a> * <a href="/A374314" title="Number of graphs of order n with exactly two Hamiltonian cycles.">A374314</a> <a href="/A055085" title="Smallest integer containing all digits in all bases from 2 to n.">A055085</a> <a href="/A354298" title="a(n) is the numerator of Sum_{k=1..n} (-1)^(k+1) / (2*k-1)!!.">A354298</a></div> <div class=sectline>Adjacent sequences: <a href="/A049360" title="Digitally balanced numbers in base 9: equal numbers of 0's, 1's, ..., 8's.">A049360</a> <a href="/A049361" title="Digitally balanced numbers in both bases 2 and 3.">A049361</a> <a href="/A049362" title="Digitally balanced numbers in bases 2, 3 and 4.">A049362</a> * <a href="/A049364" title="Smallest number that is digitally balanced in all bases 2, 3, ... n.">A049364</a> <a href="/A049365" title="Number of connected planar graphs with n nodes and even degrees (i.e., that are Eulerian).">A049365</a> <a href="/A049366" title="Number of Hamiltonian planar graphs with n nodes.">A049366</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="an exceptionally nice sequence">nice</span></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:Harvey_P._Dale">Harvey P. Dale</a></div> </div> </div> <div class=section> <div class=sectname>EXTENSIONS</div> <div class=sectbody> <div class=sectline>More terms from Ulrich Schimke (ulrschimke(AT)aol.com)</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 16 06:16 EST 2025. Contains 380893 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>