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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 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/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=%2fA006753">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 - 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> A006753 </div> <div class=seqname> Smith (or joke) numbers: composite numbers n such that sum of digits of n = sum of digits of prime factors of n (counted with multiplicity). <br><font size=-1>(Formerly M3582)</font> </div> </div> <div class=scorerefs> 85 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>4, 22, 27, 58, 85, 94, 121, 166, 202, 265, 274, 319, 346, 355, 378, 382, 391, 438, 454, 483, 517, 526, 535, 562, 576, 588, 627, 634, 636, 645, 648, 654, 663, 666, 690, 706, 728, 729, 762, 778, 825, 852, 861, 895, 913, 915, 922, 958, 985, 1086, 1111, 1165, 1219</div> <div class=seqdatalinks> (<a href="/A006753/list">list</a>; <a href="/A006753/graph">graph</a>; <a href="/search?q=A006753+-id:A006753">refs</a>; <a href="/A006753/listen">listen</a>; <a href="/history?seq=A006753">history</a>; <a href="/search?q=id:A006753&fmt=text">text</a>; <a href="/A006753/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>Of course primes also have this property, trivially.</div> <div class=sectline>a(133809) = 4937775 is the first Smith number historically: 4937775 = 3*5*5*65837 and 4+9+3+7+7+7+5 = 3+5+5+(6+5+8+3+7) = 42, Albert Wilansky coined the term Smith number when he noticed the defining property in the phone number of his brother-in-law Harold Smith: 493-7775.</div> <div class=sectline>There are 248483 7-digit Smith numbers, corresponding to US phone numbers without area codes (like 4937775). - <a href="/wiki/User:Charles_R_Greathouse_IV">Charles R Greathouse IV</a>, May 19 2013</div> <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_{k=1..<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>(a(n))} <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(n),k)), 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 class=sectline>3^3, 3^6, 3^9, 3^27 are in the sequence. - <a href="/wiki/User:Sergey_Pavlov">Sergey Pavlov</a>, Apr 01 2017</div> <div class=sectline>As mentioned by <a href="/wiki/User:Giovanni_Resta">Giovanni Resta</a>, there are no other terms of the form 3^t for 0 < t < 300000 and, probably, no other terms of such form for t >= 300000. It seems that, if there exists any other term of form 3^t with integer t, then t == 0 (mod 3) or, perhaps, t = {3^k; 2*3^k} where k is integer, k > 10. - <a href="/wiki/User:Sergey_Pavlov">Sergey Pavlov</a>, Apr 03 2017</div> </div> </div> <div class=section> <div class=sectname>REFERENCES</div> <div class=sectbody> <div class=sectline>M. Gardner, Penrose Tiles to Trapdoor Ciphers. Freeman, NY, 1989, p. 300.</div> <div class=sectline>R. K. Guy, Unsolved Problems in the Theory of Numbers, Section B49.</div> <div class=sectline>C. A. Pickover, "A Brief History of Smith Numbers" in "Wonders of Numbers: Adventures in Mathematics, Mind and Meaning", pp. 247-248, Oxford University Press, 2000.</div> <div class=sectline>J. E. Roberts, Lure of the Integers, pp. 269-270 MAA 1992.</div> <div class=sectline>N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).</div> <div class=sectline>D. D. Spencer, Key Dates in Number Theory History, Camelot Pub. Co. FL, 1995, pp. 94.</div> <div class=sectline>David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Rev. ed. 1997), p. 180.</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>T. D. Noe, <a href="/A006753/b006753.txt">Table of n, a(n) for n = 1..10000</a></div> <div class=sectline>K. S. Brown's Mathpages, <a href="http://www.mathpages.com/home/kmath007/kmath007.htm">Smith Numbers and Rhonda Numbers</a></div> <div class=sectline>C. K. Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/page.php?sort=SmithNumber">Smith number</a></div> <div class=sectline>P. J. Costello, <a href="https://web.archive.org/web/20020527191732/http://www.math.eku.edu/PJCostello/smith.htm">Smith Numbers</a></div> <div class=sectline>M. Gardner, <a href="/A005130/a005130_1.pdf">Letter to N. J. A. Sloane</a>, Jun 20 1991.</div> <div class=sectline>Ely Golden, <a href="/A006753/a006753_1.sagews.txt">General program for generating Smith number sequences</a></div> <div class=sectline>S. S. Gupta, <a href="http://www.shyamsundergupta.com/smith.htm">Smith Numbers</a></div> <div class=sectline>T. Jason, <a href="http://everything2.net/index.pl?node_id=1104442&displaytype=printable&lastnode_id=1104442">Smith number</a></div> <div class=sectline>Madras Math's Amazing Number Facts, <a href="http://www.madrasmaths.com/activities/number_facts/fact_42.html">Smith Numbers</a></div> <div class=sectline>Sham Oltikar, and Keith Wayland, <a href="http://www.jstor.org/stable/2690265">Construction of Smith Numbers</a>, Mathematics Magazine, vol. 56(1), 1983, pp. 36-37.</div> <div class=sectline>C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," <a href="http://www.zentralblatt-math.org/zmath/en/search/?q=an:0983.00008&format=complete">Zentralblatt review</a></div> <div class=sectline>Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_107.htm">Problem 107: Consecutive Smith numbers</a>, The Prime Puzzles and Problems Connection.</div> <div class=sectline>Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_108.htm">Problem 108: Methods for generating Smith numbers</a>, The Prime Puzzles and Problems Connection.</div> <div class=sectline>W. Schneider, <a href="http://web.archive.org/web/2004/www.wschnei.de/digit-related-numbers/smith-numbers.html">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="http://en.wikipedia.org/wiki/Smith_number">Smith number</a></div> <div class=sectline>A. Wilansky, <a href="http://www.jstor.org/stable/3026531">Smith numbers</a>, Two-Year Coll. Math. J., 13 (1982), p. 21.</div> <div class=sectline>A. Witno, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Witno/witno6.html">A Family of Sequences Generating Smith Numbers</a>, J. Int. Seq. 16 (2013) #13.4.6</div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>58 = 2*29; sum of digits of 58 is 13, sum of digits of 2 + sum of digits of 29 = 2+11 is also 13.</div> </div> </div> <div class=section> <div class=sectname>MAPLE</div> <div class=sectbody> <div class=sectline>q:= n-> not isprime(n) and (s-> s(n)=add(s(i[1])*i[2], i=</div> <div class=sectline> ifactors(n)[2]))(h-> add(i, i=convert(h, base, 10))):</div> <div class=sectline>select(q, [$1..2000])[]; # <a href="/wiki/User:Alois_P._Heinz">Alois P. Heinz</a>, Apr 22 2021</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>fQ[n_] := !PrimeQ@ n && n>1 && Plus @@ Flatten[ IntegerDigits@ Table[ #[[1]], {#[[2]] }] & /@ FactorInteger@ n] == Plus @@ IntegerDigits@ n; Select[ Range@ 1200, fQ]</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(Sage) is_<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> = lambda n: n > 1 and not is_prime(n) and sum(n.digits()) == sum(sum(p.digits())*m for p, m in factor(n)) # <a href="/wiki/User:D._S._McNeil">D. S. McNeil</a>, Dec 28 2010</div> <div class=sectline>(Haskell)</div> <div class=sectline>a006753 n = a006753_list !! (n-1)</div> <div class=sectline>a006753_list = [x | x <- a002808_list,</div> <div class=sectline> a007953 x == sum (map a007953 (a027746_row x))]</div> <div class=sectline>-- <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Dec 19 2011</div> <div class=sectline>(PARI) isA006753(n) = if(isprime(n), 0, my(f=factor(n)); sum(i=1, #f[, 1], sumdigits(f[i, 1])*f[i, 2]) == sumdigits(n)); \\ <a href="/wiki/User:Charles_R_Greathouse_IV">Charles R Greathouse IV</a>, Jan 03 2012; updated by <a href="/wiki/User:Max_Alekseyev">Max Alekseyev</a>, Oct 21 2016</div> <div class=sectline>(Python)</div> <div class=sectline>from sympy import factorint</div> <div class=sectline>def sd(n): return sum(map(int, str(n)))</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) == sum(sd(p)*f[p] for p in f)</div> <div class=sectline>print(list(filter(ok, range(1220)))) # <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. <a href="/A002808" title="The composite numbers: numbers n of the form x*y for x > 1 and y > 1.">A002808</a>, <a href="/A007953" title="Digital sum (i.e., sum of digits) of n; also called digsum(n).">A007953</a>, <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>, <a href="/A050218" title="Sums of digits of Smith numbers A006753.">A050218</a>, <a href="/A050224" title="1/2-Smith numbers.">A050224</a>, <a href="/A050255" title="A Diaconis-Mosteller approximation to the Birthday problem function.">A050255</a>, <a href="/A098834" title="Palindromic Smith numbers.">A098834</a>-<a href="/A098840" title="Smith triangular numbers.">A098840</a>, <a href="/A103123" title="1/4-Smith numbers.">A103123</a>-<a href="/A103126" title="5-Smith numbers.">A103126</a>, <a href="/A104166" title="Repdigit Smith numbers.">A104166</a>-<a href="/A104171" title="Reversible Smith numbers, i.e., Smith numbers whose reversal is also a Smith number.">A104171</a>, <a href="/A104390" title="2-Smith numbers.">A104390</a>, <a href="/A104391" title="3-Smith numbers.">A104391</a>, <a href="/A202387" title="Squarefree Smith numbers, cf. A006753.">A202387</a>, <a href="/A202388" title="Digital root of Smith numbers A006753.">A202388</a>.</div> <div class=sectline>Sequence in context: <a href="/A244411" title="Nonprimes n such that the product of its divisors is a palindrome.">A244411</a> <a href="/A213240" title="Numbers n such that sum of digits of n = sum of digits of n’, where n’ is the arithmetic derivative of n.">A213240</a> <a href="/A279314" title="Composite numbers n such that the sum of the prime factors of n, with multiplicity, is congruent to n (mod 9).">A279314</a> * <a href="/A098836" title="Deficient Smith numbers.">A098836</a> <a href="/A204341" title="Smith numbers with either no internal digits or all internal digits are 0.">A204341</a> <a href="/A036920" title="Composite numbers n such that digit sum of n equals digit sum of sum of its prime factors (counted with multiplicity).">A036920</a></div> <div class=sectline>Adjacent sequences: <a href="/A006750" title="Coefficients of Legendre polynomials.">A006750</a> <a href="/A006751" title="Describe the previous term! (method A - initial term is 2).">A006751</a> <a href="/A006752" title="Decimal expansion of Catalan's constant 1 - 1/9 + 1/25 - 1/49 + 1/81 - ...">A006752</a> * <a href="/A006754" title="The generalized Conway-Guy sequence w^{0}.">A006754</a> <a href="/A006755" title="The generalized Conway-Guy sequence w^{1}.">A006755</a> <a href="/A006756" title="The generalized Conway-Guy sequence w^{2}.">A006756</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>,<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:N._J._A._Sloane">N. J. A. Sloane</a></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 24 21:55 EST 2024. Contains 378084 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>