# It's getting harder and harder for my friends to like me

What is the next number in this sequence?

2, 3, 6, 7, 9, 18, 25, 27, 54, 73, 97, 129, 171, 231, 313, 327, 649, 703, ... ?

Hint 1

Examine each number for 2's. It's
Ok to play by the rules, But
See that it's fun for 71, and please
Deny high 5's to fools

Hint 2

There's no proof to be done
You can't hit a home run
You're weird, so just take a triple
Maybe one more
Til you hit the floor
Call it, for friendshipz, a cripple.

This is my first puzzle and I'm new to PSE, so feedback is much appreciated!

• Welcome to Puzzling.SE! As a heads-up, it's preferred that you hold off from including hints until about a day or so after you posted the question. On the other hand, this looks like an original, high-quality number-sequence puzzle, and those are a rarity here IMO. Feel free to take the tour and visit the help center to learn more about the site. Jun 12, 2018 at 15:04
• That said, if you go asking people questions like this, you'll get a warmer reception on PSE than amongst your current friends, so welcome onboard! Jun 12, 2018 at 15:37

## 1 Answer

These are

new maxima-so-far of the "how many steps of the Collatz iteration to 1?" function

and the next one is

871 (which takes 178 steps, versus the previous record of 170 steps starting from 703).

By way of clarification:

If we start at 2, we get [2,1]: one step. If we start at 3, we get [3, 10, 5, 16, 8, 4, 2, 1]: 7 steps, a new record. 4 goes [4,2,1], fewer than 3, so it isn't in the list. 5 goes [5,16,8,4,2,1], again no improvement. 6 goes to 3 and then follows 3's path: 8 steps, a new record. So our sequence begins 2,3,6. (Arguably it should really go 1,2,3,6.)

• dang, I guess riddle hints were not necessary at all. good to know for next time! Jun 12, 2018 at 16:45
• No, the hints were extremely helpful and I don't think I'd have solved it without them. Jun 12, 2018 at 16:51
• (In particular, the "take a triple / maybe one more" lines were ... quite informative.) Jun 12, 2018 at 16:52