Abstract
Prime partitions are partitions of integers into prime parts. In this paper, we first consider prime partitions with distinct parts. By using generating functions, we obtain some inductive formulas to calculate the number of prime partitions with distinct parts. Our formulas give two generalizations of the Euler’s formula for the integer partition case. Then, we consider general prime partitions with not necessarily distinct parts. By keeping track of the recurrence of primes in a partition and finding bijections between different prime partitions, we get some inductive formulas to calculate the number of general prime partitions. Finally, by numerical experimentation we find an approximation of some analytical formulas for the number of general prime partitions.
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Acknowledgements
The second author would like to thank Mark Gockenbach and the Department of Mathematical Sciences at Michigan Technological University for providing him financial support while doing this research. The first author would like to thank Hou Biao Li for communicating the results of [9]. The authors would like to thank Yang Yang for his help in analyzing the data in Sect. 4 and to thank Jason Gregersen for his help in simplifying the Mathematica code to calculate \(\mathcal {D}(m,n)\). The authors would also like to thank the reviewers for very useful comments on an earlier version of the paper.
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Communicated by Rosihan M. Ali.
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Sun, J., Sutela, K. Inductive Formulas Related to Prime Partitions. Bull. Malays. Math. Sci. Soc. 43, 563–579 (2020). https://doi.org/10.1007/s40840-018-0702-1
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DOI: https://doi.org/10.1007/s40840-018-0702-1
Keywords
- Euler’s formula
- Combinatorial identities
- Partitions
- Primes
- Generating functions
Mathematics Subject Classification
- 05A19