Smith number
Named after | Harold Smith ( |
---|---|
OEIS index | A006753 |
In
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 = 3 · 5 · 5 · 65837
while
- 4 + 9 + 3 + 7 + 7 + 7 + 5 = 3 + 5 + 5 + (6 + 5 + 8 + 3 + 7)
in
Mathematical definition
Let be a natural number. For base , let the function be the digit sum of in base . A natural number with prime factorisation
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
- )
Properties
W.L. McDaniel in 1987 proved that there are infinitely many Smith numbers.[1][2] The number of Smith numbers in
- 1, 6, 49, 376, 3294, 29928, 278411, 2632758, 25154060, 241882509, ... (sequence A104170 in the OEIS).
Two consecutive Smith numbers (for example, 728 and 729, or 2964 and 2965) are called Smith brothers.[3] 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[4]
Smith numbers can be constructed from factored repunits.[5][verification needed] As of 2010[update], the largest known Smith number in base 10 is
- 9 × R1031 × (104594 + 3×102297 + 1)1476 ×103913210
where R1031 is the base 10 repunit (101031 − 1)/9.[citation needed][needs update]
See also
Notes
- ^ a b Sándor & Crstici (2004) p.383
- ^
McDaniel, Wayne (1987). "The existence of infinitely many k-Smith numbers". Zbl 0608.10012.
- ^ Sándor & Crstici (2004) p.384
- ^ Shyam Sunder Gupta. "Fascinating Smith Numbers".
- ^ Hoffman (1998), pp. 205–6
References
- 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. Zbl 1079.11001.
External links
- Weisstein, Eric W. "Smith Number". MathWorld.
- Copeland, Ed. "4937775 – Smith Numbers". Numberphile. Brady Haran. Archived from the original on 2021-12-21.