If we ask which natural numbers n are not expressible as $n = ab + bc + ca$ ($0 < a < b < c$) then this is a well known open problem. Numbers not expressible in such form are called Euler's "numerus idoneus" and it is conjectured that they are finite.

If we omit the condition $a < b < c$ and assume $0 < a \leq b \leq c$ then it was proved (assuming Generalized Riemann Hypothesis) that there is only a finite number of such numbers $n$.

I am interested in the problem of expressing a prime number $p$ as $p = ab + ac + bc$ for $a \geq 1$ and $b,c \geq 2$.

Anybody knows if there is some known result related to expressing prime numbers in such form? This would yield (as a corollary) a very beautiful theorem related to spanning trees in graphs.