[img]http://yoni.valhallalegends.com/stuff/2048.png[/img]

Mathematical discussion to make this post fit for my forum.

[color=#FFFFFF][04:05:23] <[/color][color=#FFFF00]BinaryChat[/color][color=#FFFFFF]> 2^11 posts is really worth noting?
[04:05:53] [/color][color=#00FFFF]<Yoni[vL]> more worth than 2*(2*5)^3
[/color][color=#FFFFFF][04:06:17] [/color][color=#00FFFF]<Yoni[vL]> it is the first number that isn't 11-free
[/color][color=#FFFFFF][04:06:46] <[/color][color=#FFFF00]BinaryChat[/color][color=#FFFFFF]> Surely you could have used a more obscure way to say 2000.
[/color][color=#FFFFFF][04:07:22] [/color][color=#00FFFF]<Yoni[vL]> where x being n-free means in the prime factorization of x, no prime appears n or more times
[/color][color=#FFFFFF][04:07:47] [/color][color=#00FFFF]<Yoni[vL]> for example, squarefree (2-free) means it doesn't contain squares, i.e. prime factorization generates distinct primes
[/color][color=#FFFFFF][04:10:26] [/color][color=#00FFFF]<Yoni[vL]> it is known that the asymptotic density of n-free numbers is 1/zeta(n)
[/color][color=#FFFFFF][04:11:12] <[/color][color=#FFFF00][vL]Kp[/color][color=#FFFFFF]> zeta(n) = ?
[04:11:23] [/color][color=#00FFFF]<Yoni[vL]> the Riemann Zeta function
[/color][color=#FFFFFF][04:11:27] [/color][color=#00FFFF]<Yoni[vL]> the most important function in mathematics
[/color][color=#FFFFFF][04:11:37] [/color][color=#00FFFF]<Yoni[vL]> shelves full of books have been written about this function
[/color][color=#FFFFFF][04:11:51] <[/color][color=#FFFF00][vL]Kp[/color][color=#FFFFFF]> None of the shelves are within my reach atm :p
[04:12:10] [/color][color=#00FFFF]<Yoni[vL]> for an positive real s, i believe zeta(s) can be given by the formula...[/color]
[color=#FFFFFF][04:12:30] [/color][color=#00FFFF]<Yoni[vL]> zeta(s) = sum(n=1,infinity) n^-s
[/color][color=#FFFFFF][04:12:46] <[/color][color=#FFFF00][vL]Kp[/color][color=#FFFFFF]> Well, why didn't you say so in the first place? ;)
[04:12:47] [/color][color=#00FFFF]<Yoni[vL]> i.e. zeta(2) = 1/1^2 + 1/2^2 + 1/3^2 + 1/4^2 + ...
[/color][color=#FFFFFF][04:12:51] <[/color][color=#FFFF00][vL]Kp[/color][color=#FFFFFF]> yeah
[04:13:02] [/color][color=#00FFFF]<Yoni[vL]> = pi^2 / 6 (oh the beauty of calculus)
[/color][color=#FFFFFF][04:13:51] [/color][color=#00FFFF]<Yoni[vL]> for an even positive integer n, it is known that zeta(n) = pi^n / (some whole number)
[/color][color=#FFFFFF][04:14:10] [/color][color=#00FFFF]<Yoni[vL]> for an odd integer n, zeta(n) / pi^n doesn't give a rational number or a meaningful number
[/color][color=#FFFFFF][04:15:01] [/color][color=#00FFFF]<Yoni[vL]> anyway, the density of 11-free numbers is 1/zeta(11), and the density of 12-free numbers is 1/zeta(12)
[/color][color=#FFFFFF][04:15:23] [/color][color=#00FFFF]<Yoni[vL]> meaning the density of "just-12-free numbers" has to be 1/zeta(12) - 1/zeta(11)
[/color][color=#FFFFFF][04:15:27] [/color][color=#00FFFF]<Yoni[vL]> let's see what that number is
[/color][color=#FFFFFF][04:16:12] [/color][color=#00FFFF]<Yoni[vL]> hmm, i was wrong, it's not pi^n / (some whole number), it's pi^n * (some rational number). oh well, same thing[/color]
[color=#FFFFFF][04:16:36] [/color][color=#00FFFF]<Yoni[vL]> anyway, up to 10 digits, that is
[/color][color=#FFFFFF][04:16:36] [/color][color=#00FFFF]<Yoni[vL]> 0.0002479184928
[/color][color=#FFFFFF][04:16:44] [/color][color=#00FFFF]<Yoni[vL]> let's multiply it by a 1000000
[/color][color=#FFFFFF][04:16:50] [/color][color=#00FFFF]<Yoni[vL]> 247.9184928
[/color][color=#FFFFFF][04:16:56] [/color][color=#00FFFF]<Yoni[vL]> meaning, BinaryChat
[/color][color=#FFFFFF][04:17:07] [/color][color=#00FFFF]<Yoni[vL]> that less than a million, only about 248 numbers are as special as 2048.
[/color][color=#FFFFFF][04:17:37] [/color][color=#00FFFF]<Yoni[vL]> what's special about 2000? it is 4-free but not 3-free.
[/color][color=#FFFFFF][04:17:59] [/color][color=#00FFFF]<Yoni[vL]> 1/zeta(4) - 1/zeta(3) = 0.09203103034
[/color][color=#FFFFFF][04:18:12] [/color][color=#00FFFF]<Yoni[vL]> 1000000/zeta(4) - 1000000/zeta(3) = 92031.03034
[/color][color=#FFFFFF][04:18:28] [/color][color=#00FFFF]<Yoni[vL]> less than a million, there are 248 numbers as special as 2048, but over 90,000 numbers as "special" as 2000
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