In this note, we study the representation of a natural number as the sum of three natural numbers having the form
![$(n \in \mathbb{N} )$](abs/img3.gif)
,
where
a is a fixed positive integer and
![$\lfloor . \rfloor$](abs/img4.gif)
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
![$(n \in \mathbb{N} )$](abs/img3.gif)
.
And by applying Legendre's theorem, we show that every natural number is the sum of three numbers of the form
![$(n \in \mathbb{N} )$](abs/img3.gif)
and that every natural number
N ≢ 2 (mod 24)
is the sum of three numbers of the form
![$(n \in \mathbb{N} )$](abs/img3.gif)
.
On the other hand, we show that every even natural number is the sum of three numbers of the form
![$(n \in \mathbb{N} )$](abs/img3.gif)
.
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.