Smarandache Notions in number Theory and Combinatorics (Smarandache Factor Partition, Functions and Sequences)
by
Amarnath Murthy
ONGC, INDIA

Smarandache Notions in number Theory and Combinatorics ( Smarandache Factor Partition , ,Functions and Sequences) Amarnath Murthy,S.E. (E&T), Well Logging Services , Oil & Natural Gas Corporation Ltd. Chandkheda, Ahmedabad, INDIA- 380005. amarnath_murthy@yahoo.com Resume: An Electronics Engineer involved in Oil Exploration Business in a Leading Public Sector Oil and Natual gas producing Company. Key Words: Smarandache Factor Partition, Reciprocal Partition of Unity, Star Function, Star Triangle, Length and Extent of a partition. Abstract It is proposed to include 35 papers on Smarandache notions in number theory in this conference.

In paper [1] expression of unity as the sum of the reciprocal of natural numbers is explored and in this connection Smarandache Reciprocal partition of unity sets and sequences are defined. Some results and inequalities are derived and a few open problems are proposed. This also leads to the generalisation of the additive partition function defined as Smarandache Factor paritions (SFP).

In papers [2] to [10] a number of results having beautiful patterns have been derived combinatorially on the SFPs of square free numbers. A very useful idea of 'Star function' has been developed.

papers [11] to [13] deal with miscellaneous ideas related to the SFPs. In Paper [14] 'Smarandache Reciprocal Function' has been defined and an inequality on Smarandache Function is derived.

In paper [15] Some examples on 'Smarandache Lucky Methods' in Algebra , Trigonometry and calculus are given.

In paper [16] Smarandache maximum reciprocal representation function is dealt with. In papers [17] to [30] a large number of Smarandache sequences have been defined , explored and developed.

In paper [31] some new ideas on 'Least Common Multiples' are floated. In paper [32] Smarandache dual-symmetric function has been defined.Some open problems in relation to the Stirling numbers are formulated.

In paper [33] decomposition of the divisors of a natural number into pairwise co-prime sets is analysed.

In paper [34] some new conjectures and open problem on primes and divisors are given.

In paper [35] the Star Function is applied to the divisor function. References: List of papers on Smarandache notions in number theory. Papers [1] to [19] have been published in Smarandache Notions Journal Vol. 11 , No. 1-2-3, 2000. Papers. These can also be found along with papers [20] to [27] and [31] to [34] in the website http:/www.gallup.unm.edu/~smarandache.

1. Smarandache Reciprocal Partition of Unity Sets and Sequences. 2. Generalization of Partition Function, Introducing Smarandache Factor Partition.

3. A general result on smarandache Star function. 4. More results and application of the generalized Smarandache Star function. 5. Properties of Smarandache Star Triangle. 6. Smarandache Factor Partition of a typical canonical form. 7. Length / Extent of Smarandache Factor Partition. 8. Some more ideas on smarandache Factor Partitions. 9. A note on Smarandache Divisor Sequence. 10. Algorithm for listing of Smarandache Factor Partition. 11. Expansion of x raised to the power n in Smarandache Terms of permutations. 12. Miscellaneous Results and Theorems on Smarandache Terms and Factor Partitions.

13. Open Problems and conjectures on the Factor /Reciprocal Partition theory. 14. Smarandache Reciprocal function and an elementary Inequality. 15. Some funny examples of Smarandache Lucky methods in Algebra , Trigonometry and Calculus.

16. Smarandache maximum reciprocal representation function. 17. On the divisors of Smarandache Unary Sequence. 18. Some New Smarandache sequences functions and partitions. 19. Exploring some new ideas on Smarandache type sets, functions, and sequences. 20. Smarandache Function of a function and other sequence. 21. Smarandache pascal Derived sequences. 22. Depascalisation of Smarandache Pascal Derived sequences and backward extended Fibonacci Sequence.

23. Proof of the Depascalisation Theorem. 24. Smarandache Reverse Auto correlated Sequences and some Fibonacci derived Smarandache Sequences.

25. Smarandache friendly numbers and few more sequenes. 26. Smarandache Strictly Stair case sequence. 27. Smarandache Star ( Stirling) Derived Sequence. 28. Smarandache determinant Sequences. 29. Smarandache Route sequences. 30. Smarandache Geometrical partitions and sequences. 31. Some notions on Least common multiples. 32. Smarandache dual Symmetric functions and corresponding numbers of the type of Stirling numbers of the first kind.

33. Decomposition of the number of divisors of a natural number into pairwise coprime sets.

34. Some more conjectures on primes and divisors. 35. Extending the scope of some numbertheoretic functions. % include Title, Author, Address, E-mail, Resume, Keywords

Date received: November 27, 2000