  By means of Monte Carlo simulations, we study long-range site percolation on
square and simple cubic lattices with various combinations of nearest
neighbors, up to the eighth neighbors for the square lattice and the ninth
neighbors for the simple cubic lattice. We find precise thresholds for 23
systems using a single-cluster growth algorithm. Site percolation on lattices
with compact neighborhoods can be mapped to problems of lattice percolation of
extended shapes, such as disks and spheres, and the thresholds can be related
to the continuum thresholds $\eta_c$ for objects of those shapes. This mapping
implies $zp_{c} \sim 4 \eta_c = 4.51235$ in 2D and $zp_{c} \sim 8 \eta_c =
2.73512$ in 3D for large $z$ for circular and spherical neighborhoods
respectively, where $z$ is the coordination number. Fitting our data to the
form $p_c = c/(z+b)$ we find good agreement with $c = 2^d \eta_c$; the constant
$b$ represents a finite-$z$ correction term. We also study power-law fits of
the thresholds.
