 
SMARANDACHE PRIMES, SQUARES, CUBES, MPOWERS, 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 Smarandachetype functions, which is a
prime number.
For example:
Stephan [2] proved that in the Smarandache Reverse Sequence
1, 21, 321, 4321, ..., 121110987654321, ...
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 Smarandachetype 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 Smarandachetype functions, which is a perfect cube number.
And so on:
A Smarandache mPower is a term of any of the Smarandache sequences
(except, of course, the sequence involving mpowers), or a value of any
of the Smarandachetype functions, which is a perfect mpower number.
A Smarandache Palindromic Number is a term of any of the Smarandache
sequences, or a value of any of the Smarandachetype 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 Smarandachetype 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, Vol. 9, No. 12. 1998, 410.
[3] Eric W. Weisstein, "Consecutive Number Sequences" and "Smarandache
Sequences", CRC Concise Encyclopedia of Mathematics, CRC
Press, 1998, 310311 and 16611663 respectively.
 
