SOME RESULTS ON 3-CORES

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 epsilon 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 dr,(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.