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Journal of the Ramanujan Mathematical Society

Volume 35, Issue 3, September 2020  pp. 263–276.

On the treatment of partitions as factorization and further analysis

Authors:  Abhimanyu Kumar and Meenakshi Rana
Author institution:Electrical and Instrumentation Engineering Department, Thapar Institute of Engineering and Technology, Patiala 147004, Punjab, India

Summary:  The norm of an integer partition is defined as the product of the parts of that partition. This paper aims to study the norms of the integer partitions and their relation to primes. Observing that the same norm value sometimes appears for different partitions, the norm counting function r(i,n) is defined and investigated. Its generating function is proved which unravels various properties, bounds and recursive relations as well as deep connections with both multiplicative and additive partitions. Later, a formula for the norm counting function is given and the error bounds are provided. This analysis instigates the development of an asymptotic formula for the number of multiplicative partitions of n based on heuristic assumptions.

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