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]

References

↑ Pomerance, Carl B.; Crandall, Richard E. (2001). Prime Numbers: a computational perspective. Springer. pp. 78 Ex 1.86. ISBN 0-387-25282-7.
↑ Rivera, Carlos, Primes by Listing
↑ Weisstein, Eric W.. "Integer Sequence Primes". http://mathworld.wolfram.com/IntegerSequencePrimes.html. Retrieved 2011-07-28.

External links

Weisstein, Eric W.. "Smarandache–Wellin number". http://mathworld.wolfram.com/Smarandache-WellinNumber.html.
Weisstein, Eric W.. "Smarandache–Wellin prime". http://mathworld.wolfram.com/Smarandache-WellinPrime.html.
"Smarandache-Wellin number". http://planetmath.org/?op=getobj&from=objects&id={{{id}}}.
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.

v t e Prime number classes
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First 60 primes	2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281
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