  This paper examines with elementary proofs some interesting properties of
numbers in the binary quadratic form $a^2+ab+b^2$, where $a$ and $b$ are
non-negative integers. Key findings of this paper are (i) a prime number $p$
can be represented as $a^2+ab+b^2$ if and only if $p$ is of the form $6k+1$,
with the only exception of 3, (ii) any positive integer can be represented as
$a^2+ab+b^2$ if and only if its all prime factors that are not in the same form
have even exponents in the standard factorization, and (iii) all the factors of
an integer in the form $a^2+ab+b^2$, where $a$ and $b$ are positive and
relatively prime to each other, are also of the same form. A general formula
for the number of distinct representations of any positive integer in this form
is conjectured. A comparison of the results with the properties of some other
binary quadratic forms is given.
