Wandering in the World of Smarandache Numbers, by A. A. K. Majumdar, Proquest Information & Learning, Ann Arbor, MI, 2010. Available in PDF form at http://fs.gallup.unm.edu/Majumdar.pdf ISBN 9781599731247.
Like all of the really good math books, this one starts with a foundation of material that is already known and then presents new material and poses as yet unanswered questions. For example, chapter one covers some of the Smarandache sequences which are generally based on concatenation rather than formulas. Chapter two deals with the Smarandache determinant sequences, where the generation of the elements is done by taking the determinants of matrices of specific types. Chapter three covers the Smarandache function, four the Pseudo-Smarandache function and chapter five Smarandache number related triangles.
After putting forward and proving results that are known, the author extends the material with additional theorems, proofs and conjectures. To Majumdar’s credit, there is no skimping on proofs, when it can be proven, the complete proof is given.
If you have an interest in these topics and are looking for an area of research exploration, check out this book. Professor Smarandache has been an enormous source of new ideas and problems in mathematics, some of them are put forward and examined in this book. There are also a large number of different paths that can be explored, like most of number theory they are easy to state and understand, but may be very hard to prove.