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# PARTITIONS OF THE SET OF NONNEGATIVE INTEGERS WITH THE SAME REPRESENTATION FUNCTIONS

## Abstract

Let $\mathbb{N}$ be the set of all nonnegative integers. For a given set $S\subset \mathbb{N}$ the representation function $R_{S}(n)$ counts the number of solutions of the equation $n=s+s^{\prime }$ with $s<s^{\prime }$ and $s,s^{\prime }\in S$ . We obtain some results on a problem of Chen and Lev [‘Integer sets with identical representation functions’, Integers 16 (2016), Article ID A36, 4 pages] about sets $A$ and $B$ such that $A\cup B=\mathbb{N}$ , $A\cap B=r+m\mathbb{N}$ and whose representation functions coincide.

## Footnotes

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This work was supported by National Natural Science Foundation of China, grant no. 11471017.

## References

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[1] Chen, Y. G. and Lev, V. F., ‘Integer sets with identical representation functions’, Integers 16 (2016), Article ID A36, 4 pages.
[2] Dombi, G., ‘Additive properties of certain sets’, Acta Arith. 103 (2002), 137146.
[3] Kiss, S. Z. and Sándor, C., ‘Partitions of the set of nonnegative integers with the same representation functions’, Discrete Math. 340 (2017), 11541161.
[4] Kiss, S. Z. and Sándor, C., ‘On the structure of sets which has coinciding representation functions’, Preprint, 2017 arXiv:1702.04499v1.
[5] Lev, V. F., ‘Reconstructing integer sets from their representation functions’, Electron. J. Combin. 11 (2004), Article ID R78.
[6] Sándor, C., ‘Partitions of natural numbers and their representation functions’, Integers 4 (2004), Article ID A18.
[7] Tang, M., ‘Partitions of the set of natural numbers and their representation functions’, Discrete Math. 308 (2008), 26142616.
[8] Tang, M., ‘Partitions of natural numbers and their representation functions’, Chin. Ann. Math. Ser. A 37 (2016), 4146; English version, Chinese J. Contemp. Math. 37 (2016), 39–44.
[9] Yu, W. and Tang, M., ‘A note on partitions of natural numbers and their representation functions’, Integers 12 (2012), Article ID A53, 5 pages.
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