What Comes Next in This Finite Sequence?

[micromass]

I'd be extremely impressed if anybody finds this one:

70 368 744 177 664, 3 486 784 401, 1 953 125, 117 649, 121, 2197, 17, 19, 23, 29, 31, 41, 47, 59


This is a finite sequence. It can not be extended indefinitely (to my knowledge...)

 

[holomorphic]

I'd be extremely impressed if anybody finds this one:

70 368 744 177 664, 3 486 784 401, 1 953 125, 117 649, 121, 2197, 17, 19, 23, 29, 31, 41, 47, 59


This is a finite sequence. It can not be extended indefinitely (to my knowledge...)

Uhh... 71? (Guess.)

$$2^{46}, 3^{20}, 5^{9}, 7^{6}, 11^{2}, 13^{3}, ...$$
Then the ones that follow are to the first power. I don't know why the exponents would go 1,1,1,3,2,6,9,20,46...

 

[micromass]

Yes, 71 is correct. I didn't expect somebody to find this that quick...
Is there anybody that can figure out why the exponents behave this way. Hint: theory of simple groups

 

[holomorphic]

Yes, 71 is correct. I didn't expect somebody to find this that quick...
Is there anybody that can figure out why the exponents behave this way. Hint: theory of simple groups

Oh... order of the monster group. Pblackffffft. Because I use that every day.

Note: I cheated. :(

 

[micromass]

Sorry

I really like the monster group I'll keep the next ones more down-to-earth...