View source for Smith number

{{Short description|Type of composite integer}}
{{Infobox integer sequence
| named_after = Harold Smith ([[brother-in-law]] of Albert Wilansky)
| author = [[Albert Wilansky]]
| number = [[infinity]]
| first_terms = [[4 (number)|4]], [[22 (number)|22]], [[27 (number)|27]], [[58 (number)|58]], [[85 (number)|85]], [[94 (number)|94]], [[121 (number)|121]]
| OEIS = A006753
}}
In [[number theory]], a '''Smith number''' is a [[composite number]] for which, in a given [[radix|number base]], the [[digit sum|sum of its digits]] is equal to the sum of the digits in its [[prime factor]]ization in the same base.  In the case of numbers that are not [[Square-free integer|square-free]], the factorization is written without exponents, writing the repeated factor as many times as needed.

Smith numbers were named by [[Albert Wilansky]] of [[Lehigh University]], as he noticed the property in the phone number (493-7775) of his brother-in-law Harold Smith:
: 4937775&nbsp;=&nbsp;3 · 5 · 5 · 65837
while 
: 4&nbsp;+&nbsp;9&nbsp;+&nbsp;3&nbsp;+&nbsp;7&nbsp;+&nbsp;7&nbsp;+&nbsp;7&nbsp;+&nbsp;5&nbsp;= 3 + 5 + 5 + (6&nbsp;+&nbsp;5&nbsp;+&nbsp;8&nbsp;+&nbsp;3&nbsp;+&nbsp;7)
in [[base 10]].<ref name=CS383>Sándor & Crstici (2004) p.383</ref> 

==Mathematical definition==
Let <math>n</math> be a [[natural number]]. For base <math>b > 1</math>, let the function <math>F_b(n)</math> be the [[digit sum]] of <math>n</math> in base <math>b</math>. A natural number <math>n</math> with prime factorization
<math display="block">
n = \prod_{\stackrel{p \mid n,}{p\text{ prime}}} p^{v_p(n)}
</math>
is a '''Smith number''' if
<math display="block">
F_b(n) = \sum_{{\stackrel{p \mid n,}{p\text{ prime}}}} v_p(n) F_b(p).
</math>
Here the exponent <math>v_p(n)</math> is the multiplicity of <math>p</math> as a prime factor of <math>n</math> (also known as the [[p-adic valuation|''p''-adic valuation]] of <math>n</math>).

For example, in base 10, 378 = 2<sup>1</sup> · 3<sup>3</sup> · 7<sup>1</sup> is a Smith number since 3 + 7 + 8 = 2&thinsp;·&thinsp;1 + 3&thinsp;·&thinsp;3 + 7&thinsp;·&thinsp;1, and 22 = 2<sup>1</sup> · 11<sup>1</sup> is a Smith number, because 2 + 2 = 2&thinsp;·&thinsp;1 + (1 + 1)&thinsp;·&thinsp;1.

The first few Smith numbers in base 10 are
:[[4 (number)|4]], [[22 (number)|22]], [[27 (number)|27]], [[58 (number)|58]], [[85 (number)|85]], [[94 (number)|94]], [[121 (number)|121]], [[166 (number)|166]], [[202 (number)|202]], [[265 (number)|265]], [[274 (number)|274]], [[319 (number)|319]], [[346 (number)|346]], [[355 (number)|355]], [[378 (number)|378]], [[382 (number)|382]], [[391 (number)|391]], [[438 (number)|438]], [[454 (number)|454]], [[483 (number)|483]], [[517 (number)|517]], [[526 (number)|526]], [[535 (number)|535]], [[562 (number)|562]], [[576 (number)|576]], [[588 (number)|588]], [[627 (number)|627]], [[634 (number)|634]], [[636 (number)|636]], [[645 (number)|645]], [[648 (number)|648]], [[654 (number)|654]], [[663 (number)|663]], [[666 (number)|666]], [[690 (number)|690]], [[706 (number)|706]], [[728 (number)|728]], [[729 (number)|729]], [[762 (number)|762]], [[778 (number)|778]], [[825 (number)|825]], [[852 (number)|852]], [[861 (number)|861]], [[895 (number)|895]], [[913 (number)|913]], [[915 (number)|915]], [[922 (number)|922]], [[958 (number)|958]], [[985 (number)|985]]. {{OEIS|id=A006753}}

== Properties ==
W.L. McDaniel in 1987 [[mathematical proof|proved]] that there are infinitely many Smith numbers.<ref name=CS383/><ref>
{{cite journal | last = McDaniel | first = Wayne | title = The existence of infinitely many k-Smith numbers | journal = [[Fibonacci Quarterly]] | volume = 25 | issue = 1 | pages = 76–80 | date = 1987 | doi = 10.1080/00150517.1987.12429731 | zbl=0608.10012 }}</ref>
The number of Smith numbers in [[base 10]] below 10<sup>''n''</sup> for ''n'' = 1, 2, ... is given by
: 1, 6, 49, 376, 3294, 29928, 278411, 2632758, 25154060, 241882509, ... {{OEIS|id=A104170}}.

Two consecutive Smith numbers (for example, 728 and 729, or 2964 and 2965) are called '''Smith brothers'''.<ref name=CS384>Sándor & Crstici (2004) p.384</ref>  It is not known how many Smith brothers there are. The starting elements of the smallest Smith ''n''-tuple (meaning ''n'' consecutive Smith numbers) in base 10 for ''n'' = 1, 2, ... are<ref>{{Cite web |author=Shyam Sunder Gupta |url=http://www.shyamsundergupta.com/smith.htm |title=Fascinating Smith Numbers }}</ref>
: 4, 728, 73615, 4463535, 15966114, 2050918644, 164736913905, ... {{OEIS|A059754}}.

Smith numbers can be constructed from factored [[repunit]]s.<ref>Hoffman (1998), pp. 205–6</ref>{{verify source|date=May 2023}} {{as of|2010}}, the largest known Smith number in base 10  is
:9 × R<sub>1031</sub> × (10<sup>4594</sup> + 3{{e|2297}} + 1)<sup>1476</sup> {{e|3913210}}
where R<sub>1031</sub> is the base 10 [[repunit]] (10<sup>1031</sup> − 1)/9.{{cn|date=May 2023}}{{update inline|date=February 2023}}

==See also==
* [[Equidigital number]]

==Notes==
{{reflist}}

==References==
* {{cite book |author-link=Martin Gardner |first=Martin |last=Gardner |title=Penrose Tiles to Trapdoor Ciphers |year=1988 |pages=299–300}}
* {{cite book|last=Hoffman|first=Paul|title=The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth| place=New York|publisher=Hyperion|year=1998}}
* {{cite book | last1=Sándor | first1=Jozsef | last2=Crstici | first2=Borislav | title=Handbook of number theory II | url=https://archive.org/details/handbooknumberth00sand_741 | url-access=limited | location=Dordrecht | publisher=Kluwer Academic | year=2004 | isbn=1-4020-2546-7 | pages=[https://archive.org/details/handbooknumberth00sand_741/page/n33 32]–36 | zbl=1079.11001 }}

==External links==
* {{MathWorld|urlname=SmithNumber|title=Smith Number}}
* {{cite web|last=Copeland|first=Ed|author-link=Edmund Copeland|title=4937775 – Smith Numbers|url=https://www.youtube.com/watch?v=mlqAvhjxAjo |archive-url=https://ghostarchive.org/varchive/youtube/20211221/mlqAvhjxAjo |archive-date=2021-12-21 |url-status=live|work=Numberphile|date=22 December 2012 |publisher=[[Brady Haran]]}}{{cbignore}}

{{Classes of natural numbers}}
{{Divisor classes}}

[[Category:Base-dependent integer sequences]]
[[Category:Eponymous numbers in mathematics]]
[[Category:Lehigh University]]