Skip to main content Accessibility help


  • YUCHEN DING (a1) and YU-CHEN SUN (a2)


We prove that, given a positive integer $m$ , there is a sequence $\{n_{i}\}_{i=1}^{k}$ of positive integers such that

$$\begin{eqnarray}m=\frac{1}{n_{1}}+\frac{1}{n_{2}}+\cdots +\frac{1}{n_{k}}\end{eqnarray}$$
with the property that partial sums of the series $\{1/n_{i}\}_{i=1}^{k}$ do not represent other integers.


Corresponding author


Hide All
[1] Berger, M. A., Felzenbaum, A. and Fraenkel, A. S., ‘Improvements to the Newman–Znám result for disjoint covering systems’, Acta Arith. 50 (1988), 113.
[2] Chen, Y. G., ‘On integers of the forms k r - 2 n and k r 2 n + 1’, J. Number Theory 98 (2003), 310319.
[3] Erdős, P., ‘On integers of the form 2 k + p and some related problems’, Summa Brasil. Math. 2 (1950), 113123.
[4] Guo, S. and Sun, Z. W., ‘On odd covering systems with distinct moduli’, Adv. Appl. Math. 35 (2005), 182187.
[5] Pan, H. and Zhao, L. L., ‘Clique numbers of graphs and irreducible exact m-covers of the integers’, Adv. Appl. Math. 43 (2009), 2430.
[6] Porubský, Š., ‘Covering systems and generating functions’, Acta Arith. 26 (1974–1975), 223231.
[7] Porubský, Š., ‘On m times covering systems of congruences’, Acta Arith. 29 (1976), 159169.
[8] Sun, Z. W., ‘On exactly m times covers’, Israel J. Math. 77 (1992), 345348.
[9] Sun, Z. W., ‘Covering the integers by arithmetic sequences’, Acta Arith. 72 (1995), 109129.
[10] Sun, Z. W., ‘Covering the integers by arithmetic sequences II’, Trans. Amer. Math. Soc. 348 (1996), 42794320.
[11] Sun, Z. W., ‘On integers not of the form ±pa ± qb ’, Proc. Amer. Math. Soc. 128 (2000), 9971002.
[12] Zhang, M. Z., ‘A note on covering systems of residue classes’, J. Sichuan Univ. (Nat. Sci. Ed.) 26 (1989), 185188; Special Issue.
[13] Zhang, M. Z., ‘On irreducible exactly m times covering system of residue classes’, J. Sichuan Normal Univ. Nat. Sci. Ed. 28 (1991), 403408.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


MSC classification


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