  We study the asymptotic distribution of almost-prime entries of abelian
horospherical flows on {\Gamma}\SL(n,R), where {\Gamma} is either SL(n,Z) or a
cocompact lattice. In the cocompact case, we obtain a result that implies
density of almost-primes of a sufficient order, and in the space of lattices we
show the density of almost-primes in the orbits of points satisfying a certain
Diophantine condition. Along the way we give an effective equidistribution
result for arbitrary horospherical flows on the space of lattices, as well as
an effective rate for the equidistribution of arithmetic sequences of times in
abelian horospherical flows.
