Note on a Partition Function Which Assumes All Integral Values

Manvendra Tamba^1,

¹Department of Mathematics, Goa University, Taleigao Plateau, Goa, India

Cite this paper:
Manvendra Tamba. Note on a Partition Function Which Assumes All Integral Values. Turkish Journal of Analysis and Number Theory. 2014; 2(6):220-222. doi: 10.12691/tjant-2-6-5

Abstract

Let G(n) denote the number of partitions of n into distinct parts which are of the form 2m, 3m, 5m, 6m-3, 8m-3, 9m-3 or 11m-3 with parts of the form 2m, 3m, 6m-3, or 11m-3 being even in number minus the number of them with parts of the form 2m, 3m, 6m-3, or 11m-3 being odd in number. In this paper, we prove that G(n) assumes all integral values and does so infinitely often.

Keywords:
partition functions Gaussian integers Jacobi’s triple product identity

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

References:

[1]	G.E. Andrews, The Theory of Partitions, (G.-C. Rota, Ed.), Encyclopedia of Math. And its Applications, Vol. 2, Addison-Wesley, Reading, MA, 1976.
[2]	G.E. Andrews, Questions and conjectures in partition theory, Amer. Math. Monthly, 93 (1986) 708-711.
[3]	G.E. Andrews, F.J. Dyson and D. Hickerson, Partitions and indefinitequadratic forms, Invent. math., 91 (1988) 391-407.
[4]	J. Lovejoy, Lacunary Partition Functions, Math. Research Letters, 9 (2002) 191-198.
[5]	M.Tamba, On a partition function which assumes all integral values, J.Number Theory 41 (1992) 77-86.