Smarandache-Wellin prime

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:

A Smarandache–Wellin number that is also prime is called a Smarandache–Wellin prime. The first three are 2, 23 and 2357 (sequence in the OEIS). The fourth has 355 digits and ends with the digits 719.

The primes at the end of the concatenation in the Smarandache–Wellin primes are

The indices of the Smarandache–Wellin primes in the sequence of Smarandache–Wellin numbers are:

The 1429th Smarandache–Wellin number is a probable prime with 5719 digits ending in 11927, discovered by Eric W. Weisstein in 1998. 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.

The Smarandache numbers are the concatenation of the numbers 1 to n. That is:

A Smarandache prime is a Smarandache number that is also prime. However, all of the first 200000 Smarandache numbers are not prime. It is conjectured there are infinitely many Smarandache primes, but none are known as of November 2015.

Since there are no known original Smarandache primes, there are three generalizations of them to find some related primes.

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