Smarandache–Wellin number
In mathematics, a Smarandache–Wellin number is an integer that in a given base is the concatenation of the first n prime numbers written in that base. Smarandache–Wellin numbers are named after Florentin Smarandache and Paul R. Wellin.
The first decimal Smarandache–Wellin numbers are:
- 2, 23, 235, 2357, 235711, 23571113, 2357111317, 235711131719, 23571113171923, 2357111317192329, ... (sequence A019518 in the OEIS).
Smarandache–Wellin prime
A Smarandache–Wellin number that is also prime is called a Smarandache–Wellin prime. The first three are 2, 23 and 2357 (sequence A069151 in the OEIS). The fourth is 355 digits long: it is the result of concatenating the first 128 prime numbers, through 719.[1]
The primes at the end of the concatenation in the Smarandache–Wellin primes are
- 2, 3, 7, 719, 1033, 2297, 3037, 11927, ... (sequence A046284 in the OEIS).
The indices of the Smarandache–Wellin primes in the sequence of Smarandache–Wellin numbers are:
- 1, 2, 4, 128, 174, 342, 435, 1429, ... (sequence A046035 in the OEIS).
The 1429th Smarandache–Wellin number is a probable prime with 5719 digits ending in 11927, discovered by Eric W. Weisstein in 1998.[2] If it is proven prime, it will be the eighth Smarandache–Wellin prime. In March 2009, Weisstein's search showed the index of the next Smarandache–Wellin prime (if one exists) is at least 22077.[3]
See also
- Copeland–Erdős constant
- Champernowne constant, another example of a number obtained by concatenating a representation in a given base.
References
External links
- Weisstein, Eric W. "Smarandache–Wellin number". MathWorld.
- Weisstein, Eric W. "Smarandache–Wellin prime". MathWorld.
- "Smarandache-Wellin number". PlanetMath.
- List of first 54 Smarandache–Wellin numbers with factorizations
- Smarandache–Wellin primes at The Prime Glossary
- Smith, S. "A Set of Conjectures on Smarandache Sequences." Bull. Pure Appl. Sci. 15E, 101–107, 1996.
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