This kind of question has come up in the context of "smooth numbers" and their use in factoring large integers. But even in two dimensions, the real right-angled triangle has posed serious difficulties, starting with a sequence of papers by Hardy and Littlewood:

G. H. Hardy and J. E. Littlewood, Some Problems of Diophantine Approximation, in ‘’Proc. 5th Int. Congress of Mathematics” (1912), 223–229.

G. H. Hardy and J. E. Littlewood , The lattice points of a right-angled triangle, Proc. London Math. Soc. (2) 20 (1921) 15–36.

G. H. Hardy and J. E. Littlewood , The lattice points of a right-angled triangle (second memoir), Hamburg Math. Abh. 1 (1922) 212–249.

To add flesh to my comment above, suppose you wish to factor a large integer N, and use a fixed "factor base" of primes, so you attempt to write N as a product of these primes to some powers, take logarithms of both sides, divide by log N, and you have an equality of the sort that you are asking about.

For more references, there is a sequence of papers trying to solve this problem in the context of a conjecture they call the "Durfree Conjecture" about the genus of algebraic curves. They have a nice bibliography, and one of their more recent papers for this line of research that I could find for you is:

Stephen T. Yau and Letian Zhang, AN UPPER ESTIMATE OF INTEGRAL POINTS IN REAL SIMPLICES WITH AN APPLICATION TO SINGULARITY THEORY, Math. Res. Lett. 13 (2006), no. 6, 911–921.

The Ehrhart theory can bound such integer counts in real tetrahedra from above and from below, which I've also thought about a bit, but these bounds are of course always asymptotic, as Richard Stanley points out.