Neglecting coefficients of a given list

Question

Suppose I have some complicated expressions, I want to extract some terms with $\{a_1, \cdots, a_n\}$ out of the complicated expressions without considering coefficients.

For example, here is my trial.

 exp1 = a1*a2*24 + a1*a2*a3*kk*25 + a1*a2^2 + a1*a2*a3*34;
 MonomialList[exp1, {a1, a2, a3}]

Then what I have is

 {a1 a2^2, a1 a2 a3 (34 + 25 kk), 24 a1 a2}

I want to extract the $a_i$'s not the coefficients in my list, i.e., I want to eliminate (34+25kk) and the number 24 out of my results without imposing //.{34+25*kk}->1, 24->1

What I want to do is for given expressions, I want to find which combinations of $a_i$'s are present in the list form.

Answer 1

Clear["Global`*"];
exp1 = a1*a2*24 + a1*a2*a3*kk*25 + a1*a2^2 + a1*a2*a3*34;
DeleteCases[List @@ exp1, _?NumericQ | kk, {2}]

{a1 a2, a1 a2^2, a1 a2 a3, a1 a2 a3}

Answer 2

Using CoefficientRules and FromCoefficientRules:

p = a1*a2*24 + a1*a2*a3*kk*25 + a1*a2^2 + a1*a2*a3*34;

vars = {a1, a2, a3};

rep = Rule[x_, y_] /; y =!= 1 :> Rule[x, 1];

cr = CoefficientRules[#, vars] & /@ List @@ p /. rep

FromCoefficientRules[#, vars] & /@ cr

(*{a1 a2, a1 a2^2, a1 a2 a3, a1 a2 a3}*)