Wallis' product exploration

[Allen]

I think the "slowness" of the clever asymptotic behavior of \( \prod_{k=1} ^{k=n} {4k^2 \over 4k^2-1} \) can best be seen by factoring the numerator and denominator for the first few \( n\) to see what is going on:

n=2
\( 2^{6}\over 3^{2} 5^{1}\)

n=3
\( 2^{8}\over 5^{2} 7^{1}\)

n=5
\( 2^{16}\over 3^{4} 7^{2} 11^{1}\)

n=7
\( 2^{22}\over 3^{3} 5^{1} 11^{2} 13^{2}\)

n=11
\( 2^{38}\over 3^{2} 7^{2} 13^{2} 17^{2} 19^{2} 23^{1}\)

n=13
\( 2^{46}\over 3^{3} 5^{4} 7^{2} 17^{2} 19^{2} 23^{2}\)

n=17
\( 2^{64}\over 3^{6} 5^{3} 7^{1} 11^{2} 19^{2} 23^{2} 29^{2} 31^{2}\)

n=19
\( 2^{70}\over 3^{3} 5^{4} 7^{2} 11^{2} 13^{1} 23^{2} 29^{2} 31^{2} 37^{2}\)

n=23
\( 2^{84}\over 3^{6} 5^{4} 13^{2} 29^{2} 31^{2} 37^{2} 41^{2} 43^{2} 47^{1}\)

n=29
\( 2^{108}\over 3^{2} 5^{2} 7^{2} 11^{2} 17^{2} 19^{2} 31^{2} 37^{2} 41^{2} 43^{2} 47^{2} 53^{2} 59^{1}\)

Do you notice anything interesting about the relationship between \( n \) and the denominator when \( n \) is prime?

What about when it's not prime?

n=28
\( 2^{106}\over 3^{1} 5^{2} 7^{2} 11^{2} 17^{2} 19^{1} 29^{2} 31^{2} 37^{2} 41^{2} 43^{2} 47^{2} 53^{2}\)

n=29
\( 2^{108}\over 3^{2} 5^{2} 7^{2} 11^{2} 17^{2} 19^{2} 31^{2} 37^{2} 41^{2} 43^{2} 47^{2} 53^{2} 59^{1}\)

n=30
\( 2^{112}\over 7^{2} 11^{2} 17^{2} 19^{2} 31^{2} 37^{2} 41^{2} 43^{2} 47^{2} 53^{2} 59^{2} 61^{1}\)

n=31
\( 2^{114}\over 3^{2} 7^{3} 11^{2} 17^{2} 19^{2} 37^{2} 41^{2} 43^{2} 47^{2} 53^{2} 59^{2} 61^{2}\)

n=32
\( 2^{126}\over 3^{4} 5^{1} 7^{4} 11^{2} 13^{1} 17^{2} 19^{2} 37^{2} 41^{2} 43^{2} 47^{2} 53^{2} 59^{2} 61^{2}\)

n=33
\( 2^{128}\over 3^{2} 5^{2} 7^{4} 13^{2} 17^{2} 19^{2} 37^{2} 41^{2} 43^{2} 47^{2} 53^{2} 59^{2} 61^{2} 67^{1}\)

n=34
\( 2^{132}\over 3^{3} 5^{2} 7^{4} 13^{2} 19^{2} 23^{1} 37^{2} 41^{2} 43^{2} 47^{2} 53^{2} 59^{2} 61^{2} 67^{2}\)


Reminds me the highly composite binomial coefficients \( \binom{2n}{n} \).

To represent as \( \pi \) rather than \( \pi \over 2 \), one can just add 1 to the numerator like so:

\( \pi_{29} \approx \frac {2^{108+1}} {3^{2} 5^{2} 7^{2} 11^{2} 17^{2} 19^{2} 31^{2} 37^{2} 41^{2} 43^{2} 47^{2} 53^{2} 59^{1} }\)