What's the Pattern in This Number Sequence?

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The discussion revolves around identifying the next number in a complex sequence: 1, 1, 6561, 2197, 289, 21, 1, 707281. Participants analyze the sequence and deduce that the numbers can be expressed in terms of a formula involving bases and exponents. The key insight is that the exponents correspond to the bases modulo 5. The proposed formula for the k-th term is (4k+1)^{(4k+1) MOD 5}, leading to the conclusion that the next number in the sequence is 33 raised to the power of 3. This mathematical exploration highlights the intricate patterns within the number sequence.
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What's the next number?

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

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

i don't see where the 1 at the start fits in :confused:
 
tiny-tim said:
hmm …

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

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

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.
 
oh, i get it now! … it starts 11, 50 :rolleyes:
 
arildno said:
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}
 

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