Generalization Of Partition Function, Introducing 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. A number of results concerning the partition function were discovered using analytic functions by Euler, Jacobi, Hardy, Ramanujan and others. Also a number of congruence properties of the function were derived. 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.