final cog

I just raised the fact that I thought 2880 was a special number, during a conversation at work today.

The reason is: 2880 and 3360 are the sole occurrence of a saddle point in g(x) of a semi-infinite series (part of which is listed below). That's right, exactly one saddle point in the whole of infinity, and it happens at around 2880 - I don't yet know why.

There's a pint of beer, or coffee-equivalent, for anyone who can tell me what the function f(x) is, or can give me the rule, or a programme, for producing the series ... , 1008, 1440, 1680, ...

• x, f(x), g(x) = f(x)/x
• 1008, 64, 0.063492
• 1440, 84, 0.058333
• 1680, 96, 0.057143
• 2016, 112, 0.055556
• 2640, 144, 0.054545
• 2880, 144, 0.050000
• 3360, 168, 0.050000
• 3600, 176, 0.048889
• 4032, 192, 0.047619
• 5040, 192, 0.038095
• 7920, 288, 0.036364
• 9360, 336, 0.035897