This program looks for solutions ((Subscript[a, 1]), (Subscript[a, 2]), ... (Subscript[a, n])) for which the sum ((∑_(i=1))^n)(1/(Subscript[a, i]))=s holds, where s is a rational value supplied by the user. 
X[a,n,s] = {((Subscript[a, 1]), (Subscript[a, 2]), ... (Subscript[a, n]))ϵ (Z^n)|(Underscript[(∑^n), i=1])(1/(Subscript[a, i]))=s,a≤(Subscript[a, 1])≤(Subscript[a, 2])≤...≤(Subscript[a, n,])s ϵ Q}.

X[a_,0, s_] = If[s == 0, {{}}, {}];
X[a_,n_, 0] = {};
X[a_,n_,s_]:= Union@@Table[
 Prepend[#,a1]&/@X[a1, n-1, s-1/a1],
{a1,Max[⌈1/s⌉,a], ⌊n/s⌋}
]

X[2,6,2]

Y[a, n, s] is a subset of X[a, n, s] where divisibility applies. In this case, the numbers are such that (Subscript[a, i]) divides (Subscript[a, n]) for all 1≤ i≤ n-1.

Y[a_,n_,s_]:= Select[X[a,n,s], AllTrue[#[[-1]]/#[[1;;-2]], IntegerQ]&]

Y[2,6,2]

Y[5,8,3]//MatrixForm