Skip to main content Accessibility help
×
Home

ON PARTITIONS OF NONNEGATIVE INTEGERS AND REPRESENTATION FUNCTIONS

  • XIAO-HUI YAN (a1)

Abstract

Let $\mathbb{N}$ be the set of all nonnegative integers. For any set $A\subset \mathbb{N}$ , let $R(A,n)$ denote the number of representations of $n$ as $n=a+a^{\prime }$ with $a,a^{\prime }\in A$ . There is no partition $\mathbb{N}=A\cup B$ such that $R(A,n)=R(B,n)$ for all sufficiently large integers  $n$ . We prove that a partition $\mathbb{N}=A\cup B$ satisfies $|R(A,n)-R(B,n)|\leq 1$ for all nonnegative integers $n$ if and only if, for each nonnegative integer  $m$ , exactly one of $2m+1$ and $2m$ is in  $A$ .

Copyright

Footnotes

Hide All

The author is supported by the National Natural Science Foundation of China, Grant No. 11771211.

Footnotes

References

Hide All
[1] Chen, Y.-G. and Wang, B., ‘On additive properties of two special sequences’, Acta Arith. 110 (2003), 299303.
[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] Li, J.-W. and Tang, M., ‘Partitions of the set of nonnegative integers with the same representation functions’, Bull. Aust. Math. Soc. 97 (2018), 200206.
[5] Tang, M. and Chen, S.-Q., ‘On a problem of partitions of the set of nonnegative integers with the same representation functions’, Discrete Math. 341 (2018), 30753078.
[6] Tang, M. and Li, J.-W., ‘On the structure of some sets which have the same representation functions’, Period. Math. Hungar. doi:10.1007/s10998-018-0240-5.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

MSC classification

ON PARTITIONS OF NONNEGATIVE INTEGERS AND REPRESENTATION FUNCTIONS

  • XIAO-HUI YAN (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed