RECURSIVE FORMULAE FOR THE MULTIPLICATIVE PARTITION FUNCTION

For a positive integer n, let f(n) be the number of essentially different ways of writing n as a product of factors greater than 1, where two factorizations of a positive integer are said to be essentially the same if they differ only in the order of the factors. This paper gives a recursive formula for the multiplicative partition function f(n).

For convenience, we define some sets used in this paper.For a positive integer r , let M 0 r be the set of r -dimensional vectors with nonnegative integer components and M r be the set of r -dimensional vectors with nonnegative integer components not all of which are zero.The following three theorems are well known.
Theorem 4. For n ∈ M r , we have Proof.Let g(x 1 ,x 2 ,...,x r ) be the function defined in Theorem 3. Taking the ith partial logarithmic derivative of the product formula for g(x 1 ,x 2 ,...,x r ) in (4), we get Taking the ith partial derivative of the right-hand side of (4), we get Comparing the coefficients of both sides of (7), we get The theorem is proved.
For positive integers m and n, let The following properties of (m, n) are easy to obtain: From the point of view of the multiplicative partition function, Theorem 4 can be restated as the following theorem.Theorem 6. let n, t be positive integers and let p be a prime number such that p m. Then we get In [4], MacMahon presents a table of values of f (n) for those n which divide one of

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning Canfield, Erdös, and Pomerance commented that they doubted the correctness of MacMahon's figures.
From Theorem 4 we can easily be sure that Canfield, Erdös and Pomerance comment is true.

•
Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation