Max GCD Permutation Product

[Leroy Quet]

If we take a permutation [a(1),a(2),...a(m)] of
[1,2,3,...m],

we can form the product:

P = product{k=1 to m} GCD(a(k),a(k-1)),

where a(0) = a(m).

But what would the maximum P be,taken over all permutations of the first 
m positive integers?  

By hand, I get:

1, 1, 1, 2, 2, 12, 12, 48, 144,...

which corresponds with sequence A025527,
m!/LCM(1,2,3,...,m).

Now, it seems very plausible that the 2 sequences are the same.

But are they?

(I apologize if I am missing somehing simple, which seems likely.)

thanks,
Leroy Quet