Two cousins were talking about their parents' safes. "They said I can keep my comic books in their safe, if I figure out the right combination, and enter it on the keypad correctly on my first attempt," said the first cousin. "I've listed out a lot of combinations but haven't figured out if I have the right one."
"How will you know when it's right, anyway? Did they tell you some formula for the combination?" asked the second cousin.
"Yes, they said the combination has as many 3-digit numbers as possible in the range from 001 to 360, if the numbers only get bigger as the combination goes along, and if all the numbers are squarefree semiprimes with their factors uniquely chained together."
"That sounds pretty complicated! Can you give me an example of some numbers that could appear in a combination like that on a smaller range?"
"Sure. Say that only two-digit numbers in the range 1 to 36 are allowed. I figure that the sequence (10, 14, 21, 33), or (5x2, 2x7, 7x3, 3x11), is the answer."
"So, 'uniquely chained' means that whenever a prime divides some number in the combination, then it divides exactly two consecutive numbers?"
"Yes, or it only divides the first number, or only divides the last number. Like 5 and 11 in that example."
"Oh, right. ... What would the answer be if only two-digit numbers in the range 01 to 90 are allowed?"
"I figured that one out too. It's (14, 22, 33, 39, 65, 85), or (7x2, 2x11, 11x3, 3x13, 13x5, 5x17)."
"Ok, now let me think for a minute about that 1 to 360 combination ... Ok, it might take me a couple of minutes ..."
While the second cousin is thinking, can you figure out the answer?