Satellites

Elon Musk has received a contract from NASA in which he must launch N satellites into orbit in
the sequence stated on a list. The weight of the satellites is also specified on the same list.
NASA also wants all the satellites being delivered on the same day, otherwise the deal would be
terminated and transferred to Jeff Bezos (which Elon Musk cannot permit no matter the costs).
Elon Musk has at his disposal only K rocket launching platforms but no rockets, so he will have
to construct them. To manufacture K rockets as quickly as possible, they all must be identical
(only one design will be made) and as compact as possible.
What is the smallest payload limit for a rocket that would allow Elon to send the N satellites into
orbit in at most K consecutive launches?