Eligible integers represented by positive ternary quadratic forms
Assume that $f$ is a positive definite integral ternary quadratic form. Let $N_f$ denote the level of $f$. Assume that there are exactly two classes in gen$(f)$ and let $g$ be a representative of the other class. Assume further that $f$ and $g$ are in the same spinor genus. We show that if $M$ with $(M,N_f)=1$ is an eligible integer which is not square-free, then it can be represented by $f$. This generalizes Ono and Soundararajan’s 1997 result for $f=x_1^2+x_2^2+10x_3^2$, Wang and Pei’s 2001 result for $f=x_1^2+7x_2^2+7x_3^2$ and Kelley’s 2001 result for $f=x_1^2+x_2^2+7x_3^2$.