Innumber theory, aSmith number is acomposite number for which, in a givennumber base, thesum of its digits is equal to the sum of the digits in itsprime factorization in the same base. In the case of numbers that are notsquare-free, the factorization is written without exponents, writing the repeated factor as many times as needed.

Smith numbers were named byAlbert Wilansky ofLehigh University, as he noticed the property in the phone number (493-7775) of his brother-in-law Harold Smith:

4937775 = 3 · 5 · 5 · 65837

while

4 + 9 + 3 + 7 + 7 + 7 + 5 = 3 + 5 + 5 + (6 + 5 + 8 + 3 + 7)

inbase 10.^[1]

Let${\displaystyle n}$ be anatural number. For base${\displaystyle b>1}$, let the function${\displaystyle F_b(n)}$ be thedigit sum of${\displaystyle n}$ in base${\displaystyle b}$. A natural number${\displaystyle n}$ with prime factorization$${\displaystyle n=\prod _{\stackrel{p\mid n,}{p\text{ prime}}}{p}^{v_p(n)}}$$is aSmith number if$${\displaystyle F_b(n)=\sum _{\stackrel{p\mid n,}{p\text{ prime}}}{v}_p(n)F_b(p).}$$Here the exponent${\displaystyle v_p(n)}$ is the multiplicity of${\displaystyle p}$ as a prime factor of${\displaystyle n}$ (also known as thep-adic valuation of${\displaystyle n}$).

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. (sequenceA006753 in theOEIS)

W.L. McDaniel in 1987proved that there are infinitely many Smith numbers.^[1][2]The number of Smith numbers inbase 10 below 10ⁿ forn = 1, 2, ... is given by

1, 6, 49, 376, 3294, 29928, 278411, 2632758, 25154060, 241882509, ... (sequenceA104170 in theOEIS).

Two consecutive Smith numbers (for example, 728 and 729, or 2964 and 2965) are calledSmith brothers.^[3] It is not known how many Smith brothers there are. The starting elements of the smallest Smithn-tuple (meaningn consecutive Smith numbers) in base 10 forn = 1, 2, ... are^[4]

4, 728, 73615, 4463535, 15966114, 2050918644, 164736913905, ... (sequenceA059754 in theOEIS).

Smith numbers can be constructed from factoredrepunits.^{[5][verification needed]} As of 2010^[update], the largest known Smith number in base 10 is

9 × R₁₀₃₁ × (10⁴⁵⁹⁴ + 3×10²²⁹⁷ + 1)¹⁴⁷⁶×10^3913210

where R₁₀₃₁ is the base 10repunit (10¹⁰³¹ − 1)/9.^{[6][needs update]}

• Equidigital number

• Gardner, Martin (1988).Penrose Tiles to Trapdoor Ciphers. pp. 299–300.
• Hoffman, Paul (1998).The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth. New York: Hyperion.
• Sándor, Jozsef; Crstici, Borislav (2004).Handbook of number theory II. Dordrecht: Kluwer Academic. pp. 32–36.ISBN 1-4020-2546-7.Zbl 1079.11001.

• Weisstein, Eric W."Smith Number".MathWorld.
• Copeland, Ed (22 December 2012)."4937775 – Smith Numbers".Numberphile.Brady Haran.Archived from the original on 2021-12-21.

• Base-dependent integer sequences
• Eponymous numbers in mathematics
• Lehigh University

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