Smith number

Let $n$ be a natural number. For base $b\; >\; 1$, let the function $F\_b(n)$ be the digit sum of $n$ in base $b$. A natural number $n$ with prime factorization

n = \prod_{\stackrel{p \mid n,}{p\text{ prime}}} p^{v_p(n)}

is a Smith number if

F_b(n) = \sum_{{\stackrel{p \mid n,}{p\text{ prime}}}} v_p(n) F_b(p).

Here the exponent $v\_p(n)$ is the multiplicity of $p$ as a prime factor of $n$ (also known as the p-adic valuation of $n$).

For example, in base 10, 378 = 2¹ · 3³ · 7¹ is a Smith number since 3 + 7 + 8 = 2 · 1 + 3 · 3 + 7 · 1, and 22 = 2¹ · 11¹ is a Smith number, because 2 + 2 = 2 · 1 + (1 + 1) · 1.

The first few Smith numbers in base 10 are

: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. {{OEIS|id=A006753}}

W.L. McDaniel in 1987 proved that there are infinitely many Smith numbers.

{{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 }}

The number of Smith numbers in base 10 below 10ⁿ 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.Sándor & Crstici (2004) p.384 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{{Cite web |author=Shyam Sunder Gupta |url=http://www.shyamsundergupta.com/smith.htm |title=Fascinating Smith Numbers }}

: 4, 728, 73615, 4463535, 15966114, 2050918644, 164736913905, ... {{OEIS|A059754}}.

Smith numbers can be constructed from factored repunits.Hoffman (1998), pp. 205–6{{verify source|date=May 2023}} {{as of|2010}}, the largest known Smith number in base 10 is

:9 × R₁₀₃₁ × (10⁴⁵⁹⁴ + 3{{e|2297}} + 1)¹⁴⁷⁶ {{e|3913210}}

where R₁₀₃₁ is the base 10 repunit (10¹⁰³¹ − 1)/9.{{cn|date=May 2023}}{{update inline|date=February 2023}}

• Equidigital number

{{reflist}}

• {{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 }}

• {{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}}

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Category:Base-dependent integer sequences

Category:Eponymous numbers in mathematics

Category:Lehigh University