Captain Etarip, wants to plunder all the treasure islands that he can. There is exactly one island for every $n\in\mathbb N$. The $n^{\text{th}}$ treasure island contains three cities, each with $n$ golden coins. Once he begins his quest, he will visit them from poorest to richest, until the end of time.
When going on a new trip, a fleet of exactly three ships is available for rent from the local mathematician, one for each city on the island. Depending on $r^{\text{th}}$ prime number he offers as a sacrifice to the mathematician, he is given ships ranked with $r+1,r+2,r+3$ ranks.
To make his trip profitable and possible, every rank $r$ ship in fleet must satisfy the following terms:
A rank $r$ ship can at best carry less than $r$ containers and would need to carry at least one container. Each container on board needs to be filled with exactly $r$ boxes, where each box needs to be filled with exactly $r$ coins. There may be at best less than $r$ boxes on deck (ones that aren't in any of the containers).
Each container requires exactly one crew member. Each needs to be paid one golden coin.
Rogue crew members not attending containers are not allowed on board.All coins carried on a ship must either be in boxes or paid to crew members.
Otherwise, the ship would attract rival pirates and you don't want any trouble.Etarip will only target islands from which he can take all the coins.
For example, a rank $7$ ship that plans to carry $135$ coins would have two containers ($2\cdot7^2$ coins), five boxes ($5\cdot7$ coins) and two crew member ($2$ coins) to successfully transport them. But that same ship can't carry $140$ coins for example, as there are $5$ coins you can't fit anywhere on board as you are $2$ coins short for a sixth box.
A rank $7$ ship can be obtained by offering $4^{\text{th}},5^{\text{th}}$ or $6^{\text{th}}$ prime number. But one ship being able to raid a city does not guarantee that you can plunder that island. All three ships in the fleet must satisfy the conditions, as each is assigned to an equivalent city.
Given all this, can you find all the islands that Etarip will eventually plunder?
Can you prove that you have found all of them? (set of all such islands is not finite)
Will you use this information to help warn the residents of treasure islands on time, or to perhaps start your own pillaging conquest and gain advantage over Captain Etarip by predicting his next raids, is up to you. Just hurry, before he starts sailing.
Plunderable Island example:
One of the islands that Etarip will plunder is the:
$300^{\text{th}}$ island.
Which is possible to plunder because:
By sacrificing the $6^{\text{th}}$ prime number, we can take all the coins:
Class $7$ ship: $6$ containers (and crew members) and $0$ boxes. ($6\cdot50+0\cdot7=300$)
Class $8$ ship: $4$ containers (and crew members) and $5$ boxes. ($4\cdot65+5\cdot8=300$)
Class $9$ ship: $3$ containers (and crew members) and $6$ boxes. ($3\cdot82+6\cdot9=300$)