Smarandache–Wellin number

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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, ... (sequence A019518 in OEIS).

Smarandache–Wellin primes[edit]

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 OEIS). The fourth has 355 digits and ends with the digits 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 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 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[edit]


  1. ^ Pomerance, Carl B.; Crandall, Richard E. (2001). Prime Numbers: a computational perspective. Springer. pp. 78 Ex 1.86. ISBN 0-387-25282-7. 
  2. ^ Rivera, Carlos, Primes by Listing
  3. ^ Weisstein, Eric W., "Integer Sequence Primes", MathWorld. Retrieved 2011-07-28.