SMARANDACHE PRIMES, SQUARES, CUBES, M-POWERS, AND MORE GENERAL
"T"-NUMBERS
Definition:
A Smarandache Prime is a term of any of the Smarandache sequences, or a value of any of the Smarandache-type functions, which is a prime number.
For example:
Stephan [2] proved that in the Smarandache Reverse Sequence
the only Smarandache Reverse Prime is:
RSm(82) = 8281807978...121110987654321
among the first 750 terms, while Weisstein extended the search up to the first 2,739 terms, and Micha Fleuren up to the first 10,000 terms. No other prime was found so far. One conjectures that this is the only prime in the whole sequence.
Similarly one defines:
A Smarandache Square is a term of any of the Smarandache sequences (except, of course, the sequence involving squares), or a value of any of the Smarandache-type functions, which is a perfect square number.
A Smarandache Cube is a term of any of the Smarandache sequences (except, of course, the sequence involving cubes), or a value of any of the Smarandache-type functions, which is a perfect cube number.
And so on: A Smarandache m-Power is a term of any of the Smarandache sequences (except, of course, the sequence involving m-powers), or a value of any of the Smarandache-type functions, which is a perfect m-power number.
A Smarandache Palindromic Number is a term of any of the Smarandache sequences, or a value of any of the Smarandache-type functions, which is polindromic.
Generalization: Let "T" be a specific defined number (for example: perfect number, or Bell number, or almost prime number, etc.). Then: A Smarandache "T" Number is a term of any of the Smarandache sequences, or a value of any of the Smarandache-type functions, which is "T".
References:
[1] Florentin Smarandache, Only Problems, Not Solutions!, Xiquan Publ. Hse., fourth edition, 1993.
[2] Ralf W. Stephan, "Factors and Primes in Two Smarandache Sequences", Smarandache Notions Journal,
[3] Eric W. Weisstein, "Consecutive Number Sequences" and "Smarandache Sequences", CRC Concise Encyclopedia of Mathematics, CRC Press, 1998, 310-311 and 1661-1663 respectively.