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A060735 - 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>A060735 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A060735" 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=%2fA060735">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="A060735 - 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> A060735 </div> <div class=seqname> a(1)=1, a(2)=2; thereafter, a(n) is the smallest number m not yet in the sequence such that every prime that divides a(n-1) also divides m. </div> </div> <div class=scorerefs> 50 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>1, 2, 4, 6, 12, 18, 24, 30, 60, 90, 120, 150, 180, 210, 420, 630, 840, 1050, 1260, 1470, 1680, 1890, 2100, 2310, 4620, 6930, 9240, 11550, 13860, 16170, 18480, 20790, 23100, 25410, 27720, 30030, 60060, 90090, 120120, 150150, 180180, 210210</div> <div class=seqdatalinks> (<a href="/A060735/list">list</a>; <a href="/A060735/graph">graph</a>; <a href="/search?q=A060735+-id:A060735">refs</a>; <a href="/A060735/listen">listen</a>; <a href="/history?seq=A060735">history</a>; <a href="/search?q=id:A060735&fmt=text">text</a>; <a href="/A060735/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>Also, numbers k at which k / (phi(k) + 1) increases.</div> <div class=sectline>Except for the initial 1, this sequence is a primorial (<a href="/A002110" title="Primorial numbers (first definition): product of first n primes. Sometimes written prime(n)#.">A002110</a>) followed by its multiples until the next primorial, then the multiples of that primorial and so on. - Wilfredo Lopez (chakotay147138274(AT)yahoo.com), Dec 28 2006</div> <div class=sectline>a(1)=1, a(2)=2. For n >= 3, a(n) is the smallest integer > a(n-1) that is divisible by every prime which divides lcm(a(1), a(2), a(3), ..., a(n)). - <a href="/wiki/User:Leroy_Quet">Leroy Quet</a>, Feb 23 2010</div> <div class=sectline>Numbers n for which <a href="/A053589" title="Greatest primorial number (A002110) which divides n.">A053589</a>(n) = <a href="/A260188" title="Greatest primorial less than or equal to n.">A260188</a>(n), thus numbers with only one nonzero digit when written in primorial base <a href="/A049345" title="n written in primorial base.">A049345</a>. - <a href="/wiki/User:Antti_Karttunen">Antti Karttunen</a>, Aug 30 2016</div> <div class=sectline>Lexicographically earliest infinite sequence of distinct positive numbers with property that every prime that divides a(n-1) also divides a(n). - <a href="/wiki/User:N._J._A._Sloane">N. J. A. Sloane</a>, Apr 08 2022</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>Trey Deitch, <a href="/A060735/b060735.txt">Table of n, a(n) for n = 1..20000</a> (terms 1..5000 from Enrique P茅rez Herrero)</div> <div class=sectline>Michel Planat, <a href="http://arxiv.org/abs/1010.3239">Riemann hypothesis from the Dedekind psi function</a>, arXiv:1010.3239 [math.GM], 2010.</div> <div class=sectline><a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a></div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>a(1) = 1, a(n) = a(n-1) + rad(a(n-1)) with rad=<a href="/A007947" title="Largest squarefree number dividing n: the squarefree kernel of n, rad(n), radical of n.">A007947</a>, squarefree kernel. - <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Apr 10 2006</div> <div class=sectline>a(<a href="/A101301" title="The sum of the first n primes, minus n.">A101301</a>(n)+1) = <a href="/A002110" title="Primorial numbers (first definition): product of first n primes. Sometimes written prime(n)#.">A002110</a>(n). - <a href="/wiki/User:Enrique_P茅rez_Herrero">Enrique P茅rez Herrero</a>, Jun 10 2012</div> <div class=sectline>a(n) = 1 + <a href="/A343048" title="a(n) is the least number whose sum of digits in primorial base equals n.">A343048</a>(n). - <a href="/wiki/User:Antti_Karttunen">Antti Karttunen</a>, Nov 14 2024</div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>After a(2)=2 the next term must be even, so a(3)=4.</div> <div class=sectline>Then a(4) must be even so a(4) = 6.</div> <div class=sectline>Now a(5) must be a multiple of 2*3=6, so a(5)=12.</div> <div class=sectline>Then a(6)=18, a(7)=24, a(8)=30.</div> <div class=sectline>Now a(9) must be a multiple of 2*3*5 = 30, so a(9)=60. And so on.</div> </div> </div> <div class=section> <div class=sectname>MAPLE</div> <div class=sectbody> <div class=sectline>seq(seq(k*mul(ithprime(i), i=1..n-1), k=1..ithprime(n)-1), n=1..10); # <a href="/wiki/User:Vladeta_Jovovic">Vladeta Jovovic</a>, Apr 08 2004</div> <div class=sectline>a := proc(n) option remember; if n=1 then return 1 fi; a(n-1);</div> <div class=sectline>% + convert(numtheory:-factorset(%), `*`) end:</div> <div class=sectline>seq(a(n), n=1..42); # after Zumkeller, <a href="/wiki/User:Peter_Luschny">Peter Luschny</a>, Aug 30 2016</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>a = 0; Do[ b = n/(EulerPhi[ n ] + 1); If[ b > a, a = b; Print[ n ] ], {n, 1, 10^6} ]</div> <div class=sectline>f[n_] := Range[Prime[n + 1] - 1] Times @@ Prime@ Range@ n; Array[f, 7, 0] // Flatten (* <a href="/wiki/User:Robert_G._Wilson_v">Robert G. Wilson v</a>, Jul 22 2015 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(PARI) first(n)=my(v=vector(n), k=1, p=1, P=1); v[1]=1; for(i=2, n, v[i]=P*k++; if(k>p && isprime(k), p=k; P=v[i]; k=1)); v \\ <a href="/wiki/User:Charles_R_Greathouse_IV">Charles R Greathouse IV</a>, Jul 22 2015</div> <div class=sectline>(PARI) is_<a href="/A060735" title="a(1)=1, a(2)=2; thereafter, a(n) is the smallest number m not yet in the sequence such that every prime that divides a(n-1) ...">A060735</a>(n, P=1)={forprime(p=2, , n>(P*=p)||return(1); n%P&&return)} \\ <a href="/wiki/User:M._F._Hasler">M. F. Hasler</a>, Mar 14 2017</div> <div class=sectline>(Python)</div> <div class=sectline>from functools import cache;</div> <div class=sectline>from sympy import primefactors, prod</div> <div class=sectline>@cache</div> <div class=sectline>def a(n): return 1 if n == 0 else a(n-1) + prod(primefactors(a(n-1)))</div> <div class=sectline>print([a(n) for n in range(42)]) # <a href="/wiki/User:Trey_Deitch">Trey Deitch</a>, Jun 08 2024</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A000010" title="Euler totient function phi(n): count numbers <= n and prime to n.">A000010</a>, <a href="/A002110" title="Primorial numbers (first definition): product of first n primes. Sometimes written prime(n)#.">A002110</a>, <a href="/A049345" title="n written in primorial base.">A049345</a>, <a href="/A055719" title="d(n)-1 | n and n is not prime.">A055719</a>, <a href="/A101301" title="The sum of the first n primes, minus n.">A101301</a>, <a href="/A053589" title="Greatest primorial number (A002110) which divides n.">A053589</a>, <a href="/A260188" title="Greatest primorial less than or equal to n.">A260188</a>.</div> <div class=sectline>Indices of ones in <a href="/A276157" title="a(n) = {greatest primorial less than or equal to n} divided by {greatest primorial number which divides n} = A260188(n)/A053...">A276157</a> and <a href="/A267263" title="Number of nonzero digits in representation of n in primorial base.">A267263</a>.</div> <div class=sectline>One more than <a href="/A343048" title="a(n) is the least number whose sum of digits in primorial base equals n.">A343048</a>.</div> <div class=sectline>Sequence in context: <a href="/A175305" title="a(1)=1. a(n) = the smallest integer > a(n-1) where both a(n)-1 and a(n)+1 are coprime to all earlier terms of the sequence.">A175305</a> <a href="/A342702" title="Indices of records of A007015.">A342702</a> <a href="/A171923" title="Records in A031883 (values).">A171923</a> * <a href="/A181416" title="Irregular table T(n,k) = n*A178883(n,k) read by rows.">A181416</a> <a href="/A225566" title="The set of magic numbers for an idealized harmonic oscillator atomic nucleus with a biaxially deformed prolate ellipsoid sha...">A225566</a> <a href="/A273009" title="Numbers k such that (2^k + 5) / 3 is prime.">A273009</a></div> <div class=sectline>Adjacent sequences: <a href="/A060732" title="a(n) = a(n-1) + a(n - 1 minus the number of terms of a(k) == (mod 5) so far).">A060732</a> <a href="/A060733" title="a(n) = a(n-1) + a(n - 1 minus the number of terms of a(k) == (mod 6) so far).">A060733</a> <a href="/A060734" title="Natural numbers written as a square array ending in last row from left to right and rightmost column from bottom to top are ...">A060734</a> * <a href="/A060736" title="Array of square numbers read by antidiagonals in up direction.">A060736</a> <a href="/A060737" title="Number of distinct differences between consecutive divisors of n! (ordered by size).">A060737</a> <a href="/A060738" title="Number of distinct differences between consecutive divisors (ordered by increasing magnitude) of n! which are not also divis...">A060738</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:Robert_G._Wilson_v">Robert G. Wilson v</a>, Apr 23 2001</div> </div> </div> <div class=section> <div class=sectname>EXTENSIONS</div> <div class=sectbody> <div class=sectline>Definition corrected by <a href="/wiki/User:Franklin_T._Adams-Watters">Franklin T. Adams-Watters</a>, Apr 16 2009</div> <div class=sectline>Simpler definition, comments, examples from <a href="/wiki/User:N._J._A._Sloane">N. J. A. Sloane</a>, Apr 08 2022</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 1 10:10 EST 2025. Contains 380496 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>