Smarandache Reciprocal Partitions of unity sets and sequences and Smarandache Factor Partition as a generalisation of Additive Parition Function.
by
Amarnath Murthy
O.N.G.C. , Well Logging Services ,Chandkheda, Ahmedabad, INDIA- 380 005.

% include Title, Author, Address, E-mail, Resume, Keywords SMARANDACHE RECIPROCAL PARTITION OF UNITY SETS AND SEQUENCES AND SMARANDACHE FACTOR PARTITION AS A GENERALISATION OF ADDITIVE PARTITION FUNCTION.

ABSTRACT: Expression of unity as the sum of the reciprocals 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. Define Smarandache Repeatable Reciprocal partition of unity set as follows: n SRRPS(n) = { x :x = ( a1, a2, . . . , an ) where å (1/ar) = 1.} r=1 fRP(n) = order of the set SRRPS(n). We have SRRPS(1) = { (1) } , fRP(1) = 1. SRRPS(4) = { (4,4,4,4), (2,4,6,12), (2,3,7,42), (2,4,5,20), (2,6,6,6),(2,4,8,8,),(2,3,12,12), ( 4,4,3,6) , (3,3,6,6), (2,3,10,15), (2, 3,9, 18)... } fRP(4) = 14. SMARANDACHE REPEATABLE RECIPROCAL PARTITION OF UNITY SEQUENCE is defined as 1 , 1 , 3 , 14 . . . where the nth term = fRP(n) . Define SMARANDACHE DISTINCT RECIPROCAL PARTITION OF UNITY SET as follows n SDRPS(n) = { x êx = (a1, a2, . . ., an) where å (1/ar) = 1 and ai = aj Û i = j} r=1 fDP(n) = order of SDRPS(n). SDRPS(1) = { (1) } , fDP(1) = 1. SRRPS(4)={(2,4,6,12),(2,3,7,42),(2,4,5,20),(2,3,10,15) ,(2,3,9,18)} fDP(4) = 5. Smarandache Distinct Reciprocal partition of unity sequences defined as follows 1 , 0 , 1 , 5 , 72 . . . the nth term is fDP(n). SMARANDACHE FACTOR PARTITION Partition function P(n) is defined as the number of ways that a positive integer can be expressed as the sum of positive integers. Two partitions are not considered to be different if they differ only in the order of their summands. In the paper Ref.[1] “SMARANDACHE RECIPROCAL PARTITION OF UNITY SETS AND SEQUENCES” while dealing with the idea of Smarandache Reciprocal Partitions of unity we are confronted with the problem as to in how many ways a number can be expressed as the product of its divisors. Exploring this lead to the generalization of the theory of partitions.

Definition : SMARANDACHE FACTOR PARTITION FUNCTION: Let a1 , a2 , a3 , . . . ar be a set of r natural numbers and p1 , p2, p3 ,. . .pr be arbitrarily chosen distinct primes then F(a1 , a2 , a3 , . . . ar ) called the Smarandache Factor Partition of (a1 , a2 , a3 , . . . ar ) is defined as the number of ways in which the number a1 a2 a3 ar N = p1 p2 p3 . . . pr could be expressed as the product of its’ divisors. A number of theorems have been proved . A large number of beautiful patterns have been dealt with. A large number of open questions have also been proposed.

References.: Following 14 papers published in Smarandache Notions Journal Vol. 11, No. 1-2-3 Spring 2000. University of CRAIOVA, ROMANIA.

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

Amarnath Murthy 3. A general result on smarandache Star function. Amarnath Murthy 4. More results and application of the generalized Smarandache Star function.

Amarnath Murthy 5. Properties of Smarandache Star Triangle. Amarnath Murthy 6. Smarandache Factor Partition of a typical canonical form. Amarnath Murthy 7. Length / Extent of Smarandache Factor Partition. Amarnath Murthy 8. Some more ideas on smarandache Factor Partitions. Amarnath Murthy 9. A note on Smarandache Divisor Sequence. Amarnath Murthy 10. Algorithm for listing of Smarandache Factor Partition. Amarnath Murthy 11. Expansion of xn in Smarandache Terms of permutations. Amarnath Murthy 12. Miscellaneous Results and Theorems on Smarandache Terms and Factor Partitions.

Amarnath Murthy 13. Smarandache Maximum Reciprocal representation function. Amarnath Murthy 14. Open Problems and conjectures on the Factor /Reciprocal Partition theory.

Amarnath Murthy

Date received: September 28, 2000