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One nice example of deep insight is the Ramanujan Tau sequence, which comes from expanding out the product (1-q) (1-q^2) (1-q^3) all to the 24th power and multiplied by q, as q-24q^2+252q^3-...

Ramanujan noticed that the coefficients a_n have some remarkable arithmetic properties, namely the sequence is multiplicative: a_m a_n = a_{mn} when m and n are relatively prime. There's a more complicated formula when m and n have a common divisor, and he also conjectured that the size of a_p is upwards of 2*p^5.5 when p is prime.

This led to the beautiful study of modular forms and all of the above statements have profound explanations, the last of which was proven 58 years later by Deligne in 1974, as the Riemann Hypothesis for curves.

PS: there are plenty of random discoveries that can be made about the sequence, for example a_n is congruent mod 691 to the sum of the 11th powers of all divisors of n. This also has a good explanation that is now known. Lehmer conjectured in 1947 that a_n is never 0, which has been verified up to n=22798241520242687999, but is still an open question.




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