Journal of Integer Sequences, Vol. 16 (2013), Article 13.6.4 |

Department of Mathematics

University of Béjaia

Béjaia

Algeria

**Abstract:**

In this note, we study the representation of a natural number as the sum of three natural numbers having the form
,
where *a* is a fixed positive integer and
denotes the integer-part function. By applying Gauss's triangular number theorem, we show that every natural number is the sum of three numbers of the form
.
And by applying Legendre's theorem, we show that every natural number is the sum of three numbers of the form
and that every natural number
*N* ≢ 2 (mod 24)
is the sum of three numbers of the form
.
On the other hand, we show that every even natural number is the sum of three numbers of the form
.
We also propose two conjectures on the subject.

Received
January 21 2013;
revised versions received January 31 2013; May 25 2013; July 3 2013.
Published in *Journal of Integer Sequences*,
July 4 2013.

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