What's the Pattern in This Number Sequence?

[holomorphic]

What's the next number?

1, 1, 6561, 2197, 289, 21, 1, 707281, ...?

 

[tiny-tim]

hmm …

well, that's 1, 1, 9⁴, 13³, 17², 21¹, 25⁰, 29⁴ …

i don't see where the 1 at the start fits in

 

[arildno]

hmm …

well, that's 1, 1, 9⁴, 13³, 17², 21¹, 25⁰, 29⁴ …

i don't see where the 1 at the start fits in

Well, that would be:
$$1^{...}, 5^{0},9^{4},13^{3}, 17^{2}, 21^{1}, 25^{0}, 29^{4}$$
So, the next would be $33^{3}$, whatever the exponent of the first number in the sequence.

 

[tiny-tim]

oh, i get it now! … it starts 1¹, 5⁰

 

[holomorphic]

Well, that would be:
$$1^{...}, 5^{0},9^{4},13^{3}, 17^{2}, 21^{1}, 25^{0}, 29^{4}$$
So, the next would be $33^{3}$, whatever the exponent of the first number in the sequence.

Yup :)

FYI, the exponents are the bases mod 5.

So the formula for the kth term is:
$$(4k+1)^{4k+1 MOD 5}$$