Rand doesn't explicitly refer to Kant in her ouevre, however in her first essay in her book Objectivist Ethics, Rand writes:
The basic social principle of the Objectivist ethics is that just as life is an end in itself, so every living human being is and end in himself not the means to ends or the welfare of others – and that therefore, that man must live for his own sake, neither sacrificing himself to others nor sacrificing others to himself. To live for his own sake means that the achievement of his own happiness is man’s highest moral purpose.
Her premise is:
so every living human being is and end in himself not the means to ends or the welfare of others
This is looks very similar to one formulation of Kants Categorical Imperative - a philosopher whom she claims to look away from. She then jumps from this premise to what seems the logical and rational and her final conclusion:
and therefore, man must live for his own sake, neither sacrificing himself to others nor sacrificing others to himself.
This appears at first simply a restatement of the premise in ordinary language. Consider a 'human being as an end in himself'. Not only he must consider himself an end but also others must do so. Hence 'live for his own sake, neither sacrificing himself to others nor others to himself'. The second statement seems simply expository. However there is a crucial change of language tempo and emphasis.
Traditionally sacrifice is the offering of food, objects or the lives of animals to a higher purpose, in particular divine beings, as an act of propitiation or worship, and often implies ritual killing. Hence to 'sacrifice himself to others' symbolically implies setting others as higher ends to which one sacrifices himself, a killing of himself, that is negating himself entirely as an end. This is a dramatic heightening of the language of the premise: I can after all treat you as an end without negating myself as an end, or I can treat you as an end without treating your end higher than mine by that mere act.
Hence one cannot take her premise as an axiom as the second statement wasn't rationally uncovered, but rhetorically so; thus it is her second statement that is axiomatic in her system of ethics - her premise is spurious.
Kant starts from his categorical principle:
“act as if the maxim of your action were to become by your will a universal law of nature.”
This simply means, act if and only if, the maxim of your action, the law that it follows could in fact be a law binding on everyone on the Community without negating itself. For example, we do not steal, because if everyone was allowed to steal, it negates the idea of property.
From this Kant derives:
“you use humanity, whether in your own person or in the person of any other, always at the same time as an end, never merely as a means.”
This merely is the crucial difference between Rand's System & Kant's, because it allows Kant to say that one can treat an other sometimes as a means - that is have someone working for you in a factory, and thus for the time he is working being a means to my end; but not wholly as a means - that is excluding slavery. (Really one should differentiate between benevolent & tyranical work or slavery here).
Now Kant is rather famously known for giving a rationalist basis to ethics. Rand famously says that, her ethics is rationally grounded. This, on the evidence above seems like a simple polemical and ungrounded claim.
If her ethics is not rationally based, how should one understand it? Where does it fit into a typology of ethics? Has she appropriated Kants dictum without Kants rationalism? Should I understand her as being a promulgator of not rationalist ethics but virtue ethics?
