Some results on $3$-cores

Abstract:

We prove that if $u(n)$ denotes the number of representations of a nonnegative integer $n$ in the form $x^2+3y^2$ with $x,y\in \mathbb {Z}$, and $a_3(n)$ is the number of $3$-cores of $n$, then $u(12n+4)=6a_3(n)$. With the help of a classical result by L. Lorenz in 1871, we also deduce that \[ a_3(n)=d_{1,3}(3n+1)-d_{2,3}(3n+1),\] where $d_{r,3}(n)$ is the number of divisors of $n$ congruent to $r$ (mod $3$), a result proved earlier by Granville and Ono by using the theory of modular forms and by Hirschhorn and Sellers with the help of elementary generating function manipulations.