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A006753 -id:A006753 - 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>A006753 -id:A006753 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/search?q=A006753\u002b-id:A006753" 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=%2fsearch%3fq%3dA006753%2b-id%3aA006753">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="A006753 -id:A006753 - 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="A006753 -id:A006753" 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> <span dir="LTR">Search:</span> <b>a006753 -id:a006753</b> <br> <div class=resultbar> <div class=resultline></div> <div> <div class=pagelinkbox> <div> <td>Displaying 1-10 of 84 results found. </div> <div class=pagelinks> <font size=-1>page 1 <a href="/search?q=A006753%20-id%3aA006753&start=10">2</a> <a href="/search?q=A006753%20-id%3aA006753&start=20">3</a> <a href="/search?q=A006753%20-id%3aA006753&start=30">4</a> <a href="/search?q=A006753%20-id%3aA006753&start=40">5</a> <a href="/search?q=A006753%20-id%3aA006753&start=50">6</a> <a href="/search?q=A006753%20-id%3aA006753&start=60">7</a> <a href="/search?q=A006753%20-id%3aA006753&start=70">8</a> <a href="/search?q=A006753%20-id%3aA006753&start=80">9</a> </font> </div> </div> </div> <div> <font size=-1>Sort: relevance | <a href="/search?q=A006753%20-id%3aA006753&sort=references">references</a> | <a href="/search?q=A006753%20-id%3aA006753&sort=number">number</a> | <a href="/search?q=A006753%20-id%3aA006753&sort=modified">modified</a> | <a href="/search?q=A006753%20-id%3aA006753&sort=created">created</a> </font> <font size=-1>Format: long | <a href="/search?q=A006753%20-id%3aA006753&fmt=short">short</a> | <a href="/search?q=A006753%20-id%3aA006753&fmt=data">data</a> </font> </div> <div class=resultline></div> </div> <div class=space5></div> <div class=sequence> <div class=space1></div> <div class=line></div> <div class=seqhead> <div class=seqnumname> <div class=seqnum> <a href="/A050218">A050218</a> </div> <div class=seqname> Sums of digits of Smith numbers <b><a href="/A006753" title="Smith (or joke) numbers: composite numbers n such that sum of digits of n = sum of digits of prime factors of n (counted wit...">A006753</a></b>. </div> </div> <div class=scorerefs> +20<br> 4 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>4, 4, 9, 13, 13, 13, 4, 13, 4, 13, 13, 13, 13, 13, 18, 13, 13, 15, 13, 15, 13, 13, 13, 13, 18, 21, 15, 13, 15, 15, 18, 15, 15, 18, 15, 13, 17, 18, 15, 22, 15, 15, 15, 22, 13, 15, 13, 22, 22, 15, 4, 13, 13, 13, 13, 15, 17, 18, 13, 15, 15, 13, 13, 22, 17, 18, 21, 22, 13, 15</div> <div class=seqdatalinks> (<a href="/A050218/list">list</a>; <a href="/A050218/graph">graph</a>; <a href="/search?q=A050218+-id:A050218">refs</a>; <a href="/A050218/listen">listen</a>; <a href="/history?seq=A050218">history</a>; <a href="/search?q=id:A050218&fmt=text">text</a>; <a href="/A050218/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>Reinhard Zumkeller, <a href="/A050218/b050218.txt">Table of n, a(n) for n = 1..10000</a></div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SmithNumber.html">Smith Number</a>.</div> <div class=sectline>Wikipedia, <a href="http://en.wikipedia.org/wiki/Smith_number">Smith number</a>.</div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>a(n) = <a href="/A007953" title="Digital sum (i.e., sum of digits) of n; also called digsum(n).">A007953</a>(<b><a href="/A006753" title="Smith (or joke) numbers: composite numbers n such that sum of digits of n = sum of digits of prime factors of n (counted wit...">A006753</a></b>(n)).</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>d[n_]:=IntegerDigits[n]; tr[n_]:=Transpose[FactorInteger[n]]; t={}; Do[If[!PrimeQ[n]&&(x=Total[d[n]])==Total[d@tr[n][[1]]*tr[n][[2]], 2], AppendTo[t, x]], {n, 4, 1850}]; t (* <a href="/wiki/User:Jayanta_Basu">Jayanta Basu</a>, Jun 04 2013 *)</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <b><a href="/A006753" title="Smith (or joke) numbers: composite numbers n such that sum of digits of n = sum of digits of prime factors of n (counted wit...">A006753</a></b>, <a href="/A050219" title="Smaller of Smith brothers.">A050219</a>, <a href="/A050220" title="Larger of Smith brothers.">A050220</a>.</div> <div class=sectline>Cf. <a href="/A202388" title="Digital root of Smith numbers A006753.">A202388</a>, <a href="/A050223" title="Digit sums of hoax numbers A019506.">A050223</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="reluctantly accepted; please don't send more sequences like this">less</span></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:Eric_W._Weisstein">Eric W. Weisstein</a></div> </div> </div> <div class=section> <div class=sectname>EXTENSIONS</div> <div class=sectbody> <div class=sectline>Offset corrected by <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Dec 19 2011</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 class=sequence> <div class=space1></div> <div class=line></div> <div class=seqhead> <div class=seqnumname> <div class=seqnum> <a href="/A202388">A202388</a> </div> <div class=seqname> Digital root of Smith numbers <b><a href="/A006753" title="Smith (or joke) numbers: composite numbers n such that sum of digits of n = sum of digits of prime factors of n (counted wit...">A006753</a></b>. </div> </div> <div class=scorerefs> +20<br> 4 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>4, 4, 9, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 9, 4, 4, 6, 4, 6, 4, 4, 4, 4, 9, 3, 6, 4, 6, 6, 9, 6, 6, 9, 6, 4, 8, 9, 6, 4, 6, 6, 6, 4, 4, 6, 4, 4, 4, 6, 4, 4, 4, 4, 4, 6, 8, 9, 4, 6, 6, 4, 4, 4, 8, 9, 3, 4, 4, 6, 4, 9, 9, 4, 4, 9, 4, 9, 8, 9, 4, 4, 6, 9, 4, 4</div> <div class=seqdatalinks> (<a href="/A202388/list">list</a>; <a href="/A202388/graph">graph</a>; <a href="/search?q=A202388+-id:A202388">refs</a>; <a href="/A202388/listen">listen</a>; <a href="/history?seq=A202388">history</a>; <a href="/search?q=id:A202388&fmt=text">text</a>; <a href="/A202388/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>COMMENTS</div> <div class=sectbody> <div class=sectline>a(n) = <a href="/A010888" title="Digital root of n (repeatedly add the digits of n until a single digit is reached).">A010888</a>(<b><a href="/A006753" title="Smith (or joke) numbers: composite numbers n such that sum of digits of n = sum of digits of prime factors of n (counted wit...">A006753</a></b>(n)); range = {4,6,8,9}.</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>Reinhard Zumkeller, <a href="/A202388/b202388.txt">Table of n, a(n) for n = 1..10000</a></div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SmithNumber.html">Smith Number</a></div> <div class=sectline>Wikipedia, <a href="http://en.wikipedia.org/wiki/Smith_number">Smith number</a></div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>d[n_]:=IntegerDigits[n]; dr[n_]:= NestWhile[Total[d[#]]&, n, #>9&]; tr[n_]:=Transpose[FactorInteger[n]]; t1=Select[Range[4, 2.2*10^3], !PrimeQ[#]&&Total[d[#]]==Total[d@tr[#][[1]]*tr[#][[2]], 2]&]; Table[dr[n], {n, t1}] (* t1 gives Smith numbers - <a href="/wiki/User:Jayanta_Basu">Jayanta Basu</a>, Jun 04 2013 *)</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A050218" title="Sums of digits of Smith numbers A006753.">A050218</a>, <a href="/A202393" title="Digital root of Hoax Numbers A019506.">A202393</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></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Dec 19 2011</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 class=sequence> <div class=space1></div> <div class=line></div> <div class=seqhead> <div class=seqnumname> <div class=seqnum> <a href="/A059754">A059754</a> </div> <div class=seqname> Smallest term in any sequence of n consecutive Smith numbers (<b><a href="/A006753" title="Smith (or joke) numbers: composite numbers n such that sum of digits of n = sum of digits of prime factors of n (counted wit...">A006753</a></b>). </div> </div> <div class=scorerefs> +20<br> 3 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>4, 728, 73615, 4463535, 15966114, 2050918644, 164736913905, 8090674745553</div> <div class=seqdatalinks> (<a href="/A059754/list">list</a>; <a href="/A059754/graph">graph</a>; <a href="/search?q=A059754+-id:A059754">refs</a>; <a href="/A059754/listen">listen</a>; <a href="/history?seq=A059754">history</a>; <a href="/search?q=id:A059754&fmt=text">text</a>; <a href="/A059754/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>REFERENCES</div> <div class=sectbody> <div class=sectline>R. K. Guy, Unsolved Problems in Number Theory, Section B49.</div> <div class=sectline>Samuel Yates, Smith Numbers Congruent to 4 (Mod 9), Journal of Recreational Mathematics, Vol. 19(2), 1987.</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline><a href="/A059754/b059754.txt">Table of n, a(n) for n=1..8.</a></div> <div class=sectline>C. Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_107.htm">Source</a></div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>The third number is 73615 since 73615, 73616, 73617 is the first example of 3 consecutive Smith numbers.</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <b><a href="/A006753" title="Smith (or joke) numbers: composite numbers n such that sum of digits of n = sum of digits of prime factors of n (counted wit...">A006753</a></b>.</div> </div> </div> <div class=section> <div class=sectname>KEYWORD</div> <div class=sectbody> <div class=sectline><span title="next term not known, may be hard to find. please extend this sequence">hard</span>,<span title="a sequence of nonnegative numbers">nonn</span>,<span title="dependent on base used for sequence">base</span>,<span title="more terms are needed! please extend this sequence">more</span></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:Shyam_Sunder_Gupta">Shyam Sunder Gupta</a>, Feb 11 2001</div> </div> </div> <div class=section> <div class=sectname>EXTENSIONS</div> <div class=sectbody> <div class=sectline>a(6) from <a href="/wiki/User:Carlos_Rivera">Carlos Rivera</a>, Dec 19 2003</div> <div class=sectline>a(7) from Jens Kruse Andersen. <a href="/wiki/User:Max_Alekseyev">Max Alekseyev</a>, Apr 21 2010</div> <div class=sectline>a(8) from <a href="/wiki/User:Max_Alekseyev">Max Alekseyev</a>, Oct 11 2010</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 class=sequence> <div class=space1></div> <div class=line></div> <div class=seqhead> <div class=seqnumname> <div class=seqnum> <a href="/A202387">A202387</a> </div> <div class=seqname> Squarefree Smith numbers, cf. <b><a href="/A006753" title="Smith (or joke) numbers: composite numbers n such that sum of digits of n = sum of digits of prime factors of n (counted wit...">A006753</a></b>. </div> </div> <div class=scorerefs> +20<br> 3 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>22, 58, 85, 94, 166, 202, 265, 274, 319, 346, 355, 382, 391, 438, 454, 483, 517, 526, 535, 562, 627, 634, 645, 654, 663, 690, 706, 762, 778, 861, 895, 913, 915, 922, 958, 985, 1086, 1111, 1165, 1219, 1255, 1282, 1507, 1581, 1626, 1633, 1642, 1678, 1795, 1822</div> <div class=seqdatalinks> (<a href="/A202387/list">list</a>; <a href="/A202387/graph">graph</a>; <a href="/search?q=A202387+-id:A202387">refs</a>; <a href="/A202387/listen">listen</a>; <a href="/history?seq=A202387">history</a>; <a href="/search?q=id:A202387&fmt=text">text</a>; <a href="/A202387/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>COMMENTS</div> <div class=sectbody> <div class=sectline>Intersection of <b><a href="/A006753" title="Smith (or joke) numbers: composite numbers n such that sum of digits of n = sum of digits of prime factors of n (counted wit...">A006753</a></b> and <a href="/A005117" title="Squarefree numbers: numbers that are not divisible by a square greater than 1.">A005117</a>;</div> <div class=sectline>also squarefree hoax numbers: intersection of <a href="/A019506" title="Hoax numbers: composite numbers whose digit-sum equals the sum of the digit-sums of its distinct prime factors.">A019506</a> and <a href="/A005117" title="Squarefree numbers: numbers that are not divisible by a square greater than 1.">A005117</a>;</div> <div class=sectline>squarefree composite numbers m such that sum of digits of m = sum of digits of all prime factors of m.</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>Reinhard Zumkeller, <a href="/A202387/b202387.txt">Table of n, a(n) for n = 1..10000</a></div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SmithNumber.html">Smith Number</a></div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HoaxNumber.html">Hoax Number</a></div> <div class=sectline>Wikipedia, <a href="http://en.wikipedia.org/wiki/Smith_number">Smith number</a></div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(Haskell)</div> <div class=sectline>a202387 n = a202387_list !! (n-1)</div> <div class=sectline>a202387_list = [x | x <- a120944_list,</div> <div class=sectline> a007953 x == sum (map a007953 (a027746_row x))]</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A007953" title="Digital sum (i.e., sum of digits) of n; also called digsum(n).">A007953</a>, <a href="/A027746" title="Irregular triangle in which first row is 1, n-th row (n>1) gives prime factors of n with repetition.">A027746</a>, <a href="/A120944" title="Composite squarefree numbers.">A120944</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></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Dec 19 2011</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 class=sequence> <div class=space1></div> <div class=line></div> <div class=seqhead> <div class=seqnumname> <div class=seqnum> <a href="/A277580">A277580</a> </div> <div class=seqname> Numbers that are both Smith (<b><a href="/A006753" title="Smith (or joke) numbers: composite numbers n such that sum of digits of n = sum of digits of prime factors of n (counted wit...">A006753</a></b>) and Lucas-Carmichael (<a href="/A006972" title="Lucas-Carmichael numbers: squarefree composite numbers k such that p | k => p+1 | k+1.">A006972</a>). </div> </div> <div class=scorerefs> +20<br> 0 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>8164079, 8421335, 21408695, 30071327, 47324639, 77350559, 103727519, 121538879, 134151479, 202767551, 239875559, 287432495, 306871487, 466861199, 560974259, 566019167, 574342145, 592557119, 594633599, 602758079, 677913599, 832477799</div> <div class=seqdatalinks> (<a href="/A277580/list">list</a>; <a href="/A277580/graph">graph</a>; <a href="/search?q=A277580+-id:A277580">refs</a>; <a href="/A277580/listen">listen</a>; <a href="/history?seq=A277580">history</a>; <a href="/search?q=id:A277580&fmt=text">text</a>; <a href="/A277580/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><a href="/A277580/b277580.txt">Table of n, a(n) for n=1..22.</a></div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Intersection of <b><a href="/A006753" title="Smith (or joke) numbers: composite numbers n such that sum of digits of n = sum of digits of prime factors of n (counted wit...">A006753</a></b> and <a href="/A006972" title="Lucas-Carmichael numbers: squarefree composite numbers k such that p | k => p+1 | k+1.">A006972</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></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:Max_Alekseyev">Max Alekseyev</a>, Oct 21 2016</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 class=sequence> <div class=space1></div> <div class=line></div> <div class=seqhead> <div class=seqnumname> <div class=seqnum> <a href="/A357841">A357841</a> </div> <div class=seqname> Smith numbers (<b><a href="/A006753" title="Smith (or joke) numbers: composite numbers n such that sum of digits of n = sum of digits of prime factors of n (counted wit...">A006753</a></b>) for which the arithmetic derivative (<a href="/A003415" title="a(n) = n' = arithmetic derivative of n: a(0) = a(1) = 0, a(prime) = 1, a(m*n) = m*a(n) + n*a(m).">A003415</a>) is also a Smith number. </div> </div> <div class=scorerefs> +20<br> 0 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>4, 27, 85, 121, 166, 265, 517, 526, 634, 706, 778, 913, 985, 1633, 1822, 1966, 2173, 2218, 2326, 2434, 2605, 2785, 3505, 3802, 3865, 3973, 4306, 4369, 4765, 4918, 5248, 5674, 5818, 5926, 6178, 6385, 7186, 7726, 8185, 8257, 8653, 9193, 9301, 10201, 10489, 10606</div> <div class=seqdatalinks> (<a href="/A357841/list">list</a>; <a href="/A357841/graph">graph</a>; <a href="/search?q=A357841+-id:A357841">refs</a>; <a href="/A357841/listen">listen</a>; <a href="/history?seq=A357841">history</a>; <a href="/search?q=id:A357841&fmt=text">text</a>; <a href="/A357841/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><a href="/A357841/b357841.txt">Table of n, a(n) for n=1..46.</a></div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>4 = <b><a href="/A006753" title="Smith (or joke) numbers: composite numbers n such that sum of digits of n = sum of digits of prime factors of n (counted wit...">A006753</a></b>(1) and 4' = 4, so 4 is a term.</div> <div class=sectline>27 = <b><a href="/A006753" title="Smith (or joke) numbers: composite numbers n such that sum of digits of n = sum of digits of prime factors of n (counted wit...">A006753</a></b>(3) and 27' = 27, so 27 is a term.</div> <div class=sectline>85 = <b><a href="/A006753" title="Smith (or joke) numbers: composite numbers n such that sum of digits of n = sum of digits of prime factors of n (counted wit...">A006753</a></b>(5) and 85' = 22 = <b><a href="/A006753" title="Smith (or joke) numbers: composite numbers n such that sum of digits of n = sum of digits of prime factors of n (counted wit...">A006753</a></b>(2), so 85 is a term.</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>digsum[n_] := Total@IntegerDigits[n]; smithQ[n_] := CompositeQ[n] && Plus @@ (Last[#]*digsum[First@#] & /@ FactorInteger[n]) == digsum[n]; d[0] = d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[10^4], smithQ[#] && smithQ[d[#]] &] (* <a href="/wiki/User:Amiram_Eldar">Amiram Eldar</a>, Oct 21 2022 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(Magma) sm:=func<n| not IsPrime(n) and (&+Intseq(n) eq &+[ Factorisation(n)[i][2]* &+Intseq(Factorisation(n)[i][1]) : i in [1..#PrimeDivisors(n)]])>; f:=func<h |h le 1 select 0 else h*(&+[Factorisation(h)[i][2] / Factorisation(h)[i][1]: i in [1..#Factorisation(h)]])>; [n:n in [2..10700]|sm(n) and sm(Floor(f(n)))];</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A003415" title="a(n) = n' = arithmetic derivative of n: a(0) = a(1) = 0, a(prime) = 1, a(m*n) = m*a(n) + n*a(m).">A003415</a>, <b><a href="/A006753" title="Smith (or joke) numbers: composite numbers n such that sum of digits of n = sum of digits of prime factors of n (counted wit...">A006753</a></b>.</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></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:Marius_A._Burtea">Marius A. Burtea</a>, Oct 20 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 class=sequence> <div class=space1></div> <div class=line></div> <div class=seqhead> <div class=seqnumname> <div class=seqnum> <a href="/A019506">A019506</a> </div> <div class=seqname> Hoax numbers: composite numbers whose digit-sum equals the sum of the digit-sums of its distinct prime factors. </div> </div> <div class=scorerefs> +10<br> 20 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>22, 58, 84, 85, 94, 136, 160, 166, 202, 234, 250, 265, 274, 308, 319, 336, 346, 355, 361, 364, 382, 391, 424, 438, 454, 456, 476, 483, 516, 517, 526, 535, 562, 627, 634, 644, 645, 650, 654, 660, 663, 690, 702, 706, 732, 735, 762, 778, 855, 860</div> <div class=seqdatalinks> (<a href="/A019506/list">list</a>; <a href="/A019506/graph">graph</a>; <a href="/search?q=A019506+-id:A019506">refs</a>; <a href="/A019506/listen">listen</a>; <a href="/history?seq=A019506">history</a>; <a href="/search?q=id:A019506&fmt=text">text</a>; <a href="/A019506/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>Reinhard Zumkeller, <a href="/A019506/b019506.txt">Table of n, a(n) for n = 1..10000</a></div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HoaxNumber.html">Hoax Number</a></div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline><a href="/A007953" title="Digital sum (i.e., sum of digits) of n; also called digsum(n).">A007953</a>(a(n)) = sum(<a href="/A007953" title="Digital sum (i.e., sum of digits) of n; also called digsum(n).">A007953</a>(<a href="/A027748" title="Irregular triangle in which first row is 1, n-th row (n > 1) lists distinct prime factors of n.">A027748</a>(a(n),k)): k=1..<a href="/A001221" title="Number of distinct primes dividing n (also called omega(n)).">A001221</a>(a(n))) and <a href="/A066247" title="Characteristic function of composite numbers: 1 if n is composite else 0.">A066247</a>(a(n)) = 1. [<a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Dec 19 2011]</div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>22 = 2*11 and digit-sum(22) = 4 = digit-sum(2) + digit-sum(11).</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>Select[Range[2, 1000], !PrimeQ[#]&&Total[Flatten[IntegerDigits/@ Transpose[ FactorInteger[#]][[1]]]]==Total[IntegerDigits[#]]&] (* <a href="/wiki/User:Harvey_P._Dale">Harvey P. Dale</a>, Feb 24 2013 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(Haskell)</div> <div class=sectline>a019506 n = a019506_list !! (n-1)</div> <div class=sectline>a019506_list = [x | x <- a002808_list,</div> <div class=sectline> a007953 x == sum (map a007953 (a027748_row x))]</div> <div class=sectline>-- <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Dec 19 2011</div> <div class=sectline>(PARI) isok(m) = !isprime(m) && (sumdigits(m) == vecsum(apply(sumdigits, factor(m)[, 1]))); \\ <a href="/wiki/User:Michel_Marcus">Michel Marcus</a>, Feb 03 2022</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <b><a href="/A006753" title="Smith (or joke) numbers: composite numbers n such that sum of digits of n = sum of digits of prime factors of n (counted wit...">A006753</a></b>.</div> <div class=sectline>Cf. <a href="/A002808" title="The composite numbers: numbers n of the form x*y for x > 1 and y > 1.">A002808</a>, <a href="/A027748" title="Irregular triangle in which first row is 1, n-th row (n > 1) lists distinct prime factors of n.">A027748</a>, <a href="/A001221" title="Number of distinct primes dividing n (also called omega(n)).">A001221</a>, <a href="/A007953" title="Digital sum (i.e., sum of digits) of n; also called digsum(n).">A007953</a>, <a href="/A050223" title="Digit sums of hoax numbers A019506.">A050223</a>, <a href="/A202393" title="Digital root of Hoax Numbers A019506.">A202393</a>, <a href="/A202387" title="Squarefree Smith numbers, cf. A006753.">A202387</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></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline>Mario Velucchi (mathchess(AT)velucchi.it)</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 class=sequence> <div class=space1></div> <div class=line></div> <div class=seqhead> <div class=seqnumname> <div class=seqnum> <a href="/A178193">A178193</a> </div> <div class=seqname> Smith numbers of order 4. </div> </div> <div class=scorerefs> +10<br> 14 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>3777, 7773, 17418, 30777, 53921, 66111, 97731, 111916, 119217, 122519, 128131, 133195, 135488, 138878, 145229, 178814, 180174, 198581, 257376, 269636, 281179, 296396, 317686, 358256, 362996, 366514, 394114, 435777, 457377, 469552, 475856, 502960, 513833</div> <div class=seqdatalinks> (<a href="/A178193/list">list</a>; <a href="/A178193/graph">graph</a>; <a href="/search?q=A178193+-id:A178193">refs</a>; <a href="/A178193/listen">listen</a>; <a href="/history?seq=A178193">history</a>; <a href="/search?q=id:A178193&fmt=text">text</a>; <a href="/A178193/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>COMMENTS</div> <div class=sectbody> <div class=sectline>Composite numbers n not in <a href="/A176670" title="Composite numbers having the same digits as their prime factors (with multiplicity), excluding zero digits.">A176670</a> such that the sum of the 4th power of the digits of n equals the sum of the 4th power of the digits of the prime factors of n (with multiplicity). <a href="/A176670" title="Composite numbers having the same digits as their prime factors (with multiplicity), excluding zero digits.">A176670</a> lists composite numbers having the same digits as their prime factors (with multiplicity), excluding zero digits.</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>Donovan Johnson, <a href="/A178193/b178193.txt">Table of n, a(n) for n = 1..1000</a></div> <div class=sectline>Patrick Costello, <a href="https://www.fq.math.ca/Scanned/40-4/costello.pdf">A new largest Smith number</a>, Fibonacci Quarterly 40(4) (2002), 369-371.</div> <div class=sectline>Underwood Dudley, <a href="https://www.jstor.org/stable/2690561">Smith numbers</a>, Mathematics Magazine 67(1) (1994), 62-65.</div> <div class=sectline>S. S. Gupta, <a href="http://www.appliedprobability.org/data/files/MS%20issues/Vol37_No1.pdf">Smith Numbers</a>, Mathematical Spectrum 37(1) (2004/5), 27-29.</div> <div class=sectline>S. S. Gupta, <a href="http://www.shyamsundergupta.com/smith.htm">Smith Numbers</a>.</div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SmithNumber.html">Smith number</a>.</div> <div class=sectline>Wikipedia, <a href="https://en.wikipedia.org/wiki/Smith_number">Smith number</a>.</div> <div class=sectline>A. Wilansky, <a href="https://www.jstor.org/stable/3026531">Smith Numbers</a>, Two-Year College Math. J. 13(1) (1982), p. 21.</div> <div class=sectline>Amin Witno, <a href="https://projecteuclid.org/euclid.mjms/1312233139">Another simple construction of Smith numbers</a>, Missouri J. Math. Sci. 22(2) (2010), 97-101.</div> <div class=sectline>Amin Witno, <a href="http://thaijmath.in.cmu.ac.th/index.php/thaijmath/article/view/952">Smith multiples of a class of primes with small digital sum</a>, Thai Journal of Mathematics 14(2) (2016), 491-495.</div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>3777 = 3*1259 is composite; sum of 4th power of the digits is 3^4 + 7^4 + 7^4 + 7^4 = 7284. Sum of 4th power of the digits of the prime factors 3, 1259 is 3^4 + 1^4 + 2^4 + 5^4 + 9^4 = 7284. The sums are equal, so 3777 is in the sequence.</div> <div class=sectline>17418 = 2*3*2903 is composite; sum of 4th power of the digits is 1^4 + 7^4 + 4^4 + 1^4 + 8^4 = 6755. Sum of 4th power of the digits of the prime factors 2, 3, 2903 is 2^4 + 3^4 + 2^4 + 9^4 + 0^4 + 3^4 = 6755. The sums are equal, so 17418 is in the sequence.</div> <div class=sectline>269636 = 2*2*67409 is composite; sum of 4th power of the digits is 2^4 + 6^4 + 9^4 + 6^4 + 3^4 + 6^4 = 10546. Sum of 4th power of the digits of the prime factors 2, 2, 67409 (with multiplicity) is 2^4 + 2^4 + 6^4 + 7^4 + 4^4 + 0^4 + 9^4 = 10546. The sums are equal, so 269636 is in the sequence.</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>fQ[n_] := Block[{id = Sort@ IntegerDigits@ n, fid = Sort@ Flatten[ IntegerDigits@ Table[#[[1]], {#[[2]]}] & /@ FactorInteger@ n]}, While[ id[[1]] == 0, id = Drop[id, 1]]; While[ fid[[1]] == 0, fid = Drop[fid, 1]]; id != fid && Plus @@ (id^4) == Plus @@ (fid^4)]; k = 1; lst = {}; While[k < 10^6, If[f Q@ k, AppendTo[lst, k]; Print@ k]; k++]; lst</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <b><a href="/A006753" title="Smith (or joke) numbers: composite numbers n such that sum of digits of n = sum of digits of prime factors of n (counted wit...">A006753</a></b> (Smith numbers), <a href="/A176670" title="Composite numbers having the same digits as their prime factors (with multiplicity), excluding zero digits.">A176670</a>, <a href="/A174460" title="Smith numbers of order 2.">A174460</a>, <a href="/A178213" title="Smith numbers of order 3.">A178213</a>, <a href="/A178203" title="Smith numbers of order 5; composite numbers n such that sum of digits^5 equal sum of digits^5 of its prime factors without t...">A178203</a>, <a href="/A178204" title="Smith numbers of order 6; composite numbers n such that sum of digits^6 equal sum of digits^6 of its prime factors without t...">A178204</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></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:Paul_Weisenhorn">Paul Weisenhorn</a>, Dec 19 2010</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 class=sequence> <div class=space1></div> <div class=line></div> <div class=seqhead> <div class=seqnumname> <div class=seqnum> <a href="/A176670">A176670</a> </div> <div class=seqname> Composite numbers having the same digits as their prime factors (with multiplicity), excluding zero digits. </div> </div> <div class=scorerefs> +10<br> 12 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>1111, 1255, 12955, 17482, 25105, 28174, 51295, 81229, 91365, 100255, 101299, 105295, 107329, 110191, 110317, 117067, 124483, 127417, 129595, 132565, 137281, 145273, 146137, 149782, 163797, 171735, 174082, 174298, 174793, 174982, 193117, 208174, 210181, 217894</div> <div class=seqdatalinks> (<a href="/A176670/list">list</a>; <a href="/A176670/graph">graph</a>; <a href="/search?q=A176670+-id:A176670">refs</a>; <a href="/A176670/listen">listen</a>; <a href="/history?seq=A176670">history</a>; <a href="/search?q=id:A176670&fmt=text">text</a>; <a href="/A176670/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>COMMENTS</div> <div class=sectbody> <div class=sectline>Subsequence of <b><a href="/A006753" title="Smith (or joke) numbers: composite numbers n such that sum of digits of n = sum of digits of prime factors of n (counted wit...">A006753</a></b> (Smith numbers).</div> <div class=sectline>These numbers still need a better name. - <a href="/wiki/User:Ely_Golden">Ely Golden</a>, Dec 25 2016</div> <div class=sectline>Terms of this sequence never have more zero digits than their prime factors. - <a href="/wiki/User:Ely_Golden">Ely Golden</a>, Jan 10 2017</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>Ely Golden, <a href="/A176670/b176670.txt">Table of n, a(n) for n = 1..10000</a> [Terms 1 through 2113 were computed by Paul Weisenhorn; and terms 2114 to 10000 by Ely Golden, Nov 30 2016]</div> <div class=sectline>Ely Golden, <a href="/A176670/a176670_1.sagews.txt">Smith number sequence generator optimized for A176670</a></div> <div class=sectline>Ely Golden, <a href="/A280827/a280827.txt">Proof that A280827(n)>=0 for all n>1</a></div> <div class=sectline>Eric W. Weisstein, <a href="http://mathworld.wolfram.com/SmithNumber.html">Smith Number</a></div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>n = 25105 = 5*5021; both n and the factorization of n have digits 1, 2, 5, 5; sorted and excluding zeros.</div> <div class=sectline>n = 110191 = 101*1091; both n and the factorization of n have digits 1, 1, 1, 1, 9; sorted and excluding zeros.</div> <div class=sectline>n = 171735 = 3*5*107*107; both n and the factorization of n have digits 1, 1, 3, 5, 7, 7; sorted and excluding zeros.</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>fQ[n_] := Block[{id = Sort@ IntegerDigits@ n, s = Sort@ Flatten[ IntegerDigits@ Table[ #[[1]], {#[[2]]}] & /@ FactorInteger@ n]}, While[ id[[1]] == 0, id = Drop[id, 1]]; While[ s[[1]] == 0, s = Drop[s, 1]]; n > 1 && ! PrimeQ@ n && s == id]; Select[ Range@ 200000, fQ]</div> <div class=sectline>Select[Range[2*10^5], Function[n, And[CompositeQ@ n, Sort@ DeleteCases[#, 0] &@ IntegerDigits@ n == Sort@ DeleteCases[#, 0] &@ Flatten@ Map[IntegerDigits@ ConstantArray[#1, #2] & @@ # &, FactorInteger@ n]]]] (* <a href="/wiki/User:Michael_De_Vlieger">Michael De Vlieger</a>, Dec 10 2016 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(Python)</div> <div class=sectline>from sympy import factorint, flatten</div> <div class=sectline>def sd(n): return sorted(str(n).replace('0', ''))</div> <div class=sectline>def ok(n):</div> <div class=sectline> f = factorint(n)</div> <div class=sectline> return sum(f[p] for p in f) > 1 and sd(n) == sorted(flatten(sd(p)*f[p] for p in f))</div> <div class=sectline>print(list(filter(ok, range(220000)))) # <a href="/wiki/User:Michael_S._Branicky">Michael S. Branicky</a>, Apr 22 2021</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <b><a href="/A006753" title="Smith (or joke) numbers: composite numbers n such that sum of digits of n = sum of digits of prime factors of n (counted wit...">A006753</a></b>.</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></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:Paul_Weisenhorn">Paul Weisenhorn</a>, Apr 23 2010</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 class=sequence> <div class=space1></div> <div class=line></div> <div class=seqhead> <div class=seqnumname> <div class=seqnum> <a href="/A278909">A278909</a> </div> <div class=seqname> Binary Smith numbers: composite numbers n such that sum of bits of n = sum of bits of prime factors of n (counted with multiplicity). </div> </div> <div class=scorerefs> +10<br> 12 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>15, 51, 55, 85, 125, 159, 185, 190, 205, 215, 222, 238, 246, 249, 253, 287, 303, 319, 374, 407, 438, 442, 469, 471, 475, 489, 494, 501, 507, 591, 623, 639, 670, 679, 687, 699, 730, 745, 755, 763, 765, 771, 799, 807, 822, 830, 843, 867, 890, 893, 917, 923, 925, 935, 939, 951, 970, 973, 979, 986, 989, 995, 1010, 1015, 1017, 1020, 1023, 1135, 1167, 1203, 1243</div> <div class=seqdatalinks> (<a href="/A278909/list">list</a>; <a href="/A278909/graph">graph</a>; <a href="/search?q=A278909+-id:A278909">refs</a>; <a href="/A278909/listen">listen</a>; <a href="/history?seq=A278909">history</a>; <a href="/search?q=id:A278909&fmt=text">text</a>; <a href="/A278909/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>COMMENTS</div> <div class=sectbody> <div class=sectline>Binary equivalent of <b><a href="/A006753" title="Smith (or joke) numbers: composite numbers n such that sum of digits of n = sum of digits of prime factors of n (counted wit...">A006753</a></b> as well as <a href="/A176670" title="Composite numbers having the same digits as their prime factors (with multiplicity), excluding zero digits.">A176670</a>. (Since bits can only be 0 or 1, having equal sums of bits is logically equivalent to having the same nonzero bits.)</div> <div class=sectline>There are 615 terms up to 10^4, 6412 up to 10^5, 66369 up to 10^6, 630106 up to 10^7, 6268949 up to 10^8, 62159262 up to 10^9, and 596587090 up to 10^10. - <a href="/wiki/User:Charles_R_Greathouse_IV">Charles R Greathouse IV</a>, Dec 09 2016</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>Ely Golden, <a href="/A278909/b278909.txt">Table of n, a(n) for n = 1..10000</a></div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>a(1) = 15, as 15 (1111) in binary has the same number of 1 bits as its prime factors (11 and 101).</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>Select[Range@ 1250, And[CompositeQ@ #, DigitCount[#, 2, 1] = Total@ Flatten@ Apply[DigitCount[#, 2, 1] & /@ ConstantArray[#1, #2] &, FactorInteger@ #, 1]] &] (* <a href="/wiki/User:Michael_De_Vlieger">Michael De Vlieger</a>, Dec 02 2016 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(SageMath)</div> <div class=sectline>def numfactorbits(x):</div> <div class=sectline> if(x<2):</div> <div class=sectline> return 0;</div> <div class=sectline> s=0;</div> <div class=sectline> f=list(factor(x));</div> <div class=sectline> #ensures inequality of numfactorbits(x) and bin(x).count("1") if x is prime</div> <div class=sectline> if((len(f)==1)&(f[0][1]==1)):</div> <div class=sectline> return 0;</div> <div class=sectline> for c in range(len(f)):</div> <div class=sectline> s+=bin(f[c][0]).count("1")*f[c][1]</div> <div class=sectline> return s;</div> <div class=sectline>counter=2</div> <div class=sectline>index=1</div> <div class=sectline>while(index<=10000):</div> <div class=sectline> if(numfactorbits(counter)==bin(counter).count("1")):</div> <div class=sectline> print(str(index)+" "+str(counter))</div> <div class=sectline> index+=1;</div> <div class=sectline> counter+=1;</div> <div class=sectline>(PARI) is(n) = my(f=factor(n)[, 1]~, expo=factor(n)[, 2]~, v=[], s=0); for(k=1, #f, while(expo[k] > 0, expo[k]--; v=concat(v, f[k]))); for(k=1, #v, v[k]=binary(v[k])); my(w=[]); for(y=1, #v, w=concat(w, v[y])); if(vecsum(w)==vecsum(binary(n)), return(1), return(0))</div> <div class=sectline>terms(n) = my(i=0); forcomposite(c=1, , if(is(c), print1(c, ", "); i++; if(i==n, break)))</div> <div class=sectline>/* Print initial 70 terms as follows: */</div> <div class=sectline>terms(70) \\ <a href="/wiki/User:Felix_Fr枚hlich">Felix Fr枚hlich</a>, Dec 01 2016</div> <div class=sectline>(PARI) is(n)=my(f=factor(n), t=#f~); (t>1 || (t==1 && f[1, 2]>1)) && hammingweight(n)==sum(i=1, t, hammingweight(f[i, 1])*f[i, 2]) \\ <a href="/wiki/User:Charles_R_Greathouse_IV">Charles R Greathouse IV</a>, Dec 02 2016</div> <div class=sectline>(Python)</div> <div class=sectline>from sympy import factorint</div> <div class=sectline>def sbd(n): return bin(n).count('1')</div> <div class=sectline>def ok(n):</div> <div class=sectline> f = factorint(n)</div> <div class=sectline> return sum(f[p] for p in f) > 1 and sbd(n) == sum(sbd(p)*f[p] for p in f)</div> <div class=sectline>print(list(filter(ok, range(1244)))) # <a href="/wiki/User:Michael_S._Branicky">Michael S. Branicky</a>, Apr 22 2021</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <b><a href="/A006753" title="Smith (or joke) numbers: composite numbers n such that sum of digits of n = sum of digits of prime factors of n (counted wit...">A006753</a></b>, <a href="/A176670" title="Composite numbers having the same digits as their prime factors (with multiplicity), excluding zero digits.">A176670</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="it is very easy to produce terms of sequence">easy</span></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:Ely_Golden">Ely Golden</a>, Nov 30 2016</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 class=resultbar> <div class=resultline></div> <div> <div class=pagelinkbox> <div></div> <div class=pagelinks> <font size=-1>page 1 <a href="/search?q=A006753%20-id%3aA006753&start=10">2</a> <a href="/search?q=A006753%20-id%3aA006753&start=20">3</a> <a href="/search?q=A006753%20-id%3aA006753&start=30">4</a> <a href="/search?q=A006753%20-id%3aA006753&start=40">5</a> <a href="/search?q=A006753%20-id%3aA006753&start=50">6</a> <a href="/search?q=A006753%20-id%3aA006753&start=60">7</a> <a href="/search?q=A006753%20-id%3aA006753&start=70">8</a> <a href="/search?q=A006753%20-id%3aA006753&start=80">9</a> </font> </div> </div> </div> <div class=resultline></div> </div> <center><p><font size=-1> Search completed in 0.031 seconds </font></p></center> </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 24 23:08 EST 2024. 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