I'm trying to define a new [silly] calendar, because there aren't enough of them yet.
My calendar, so far, is specified:
epoch (orthodox) is the moment the leading edges of the dinosaur-killing asteroid touched the surface of the earth.
As this is somewhat difficult to define, for practical purposes we'll use:
epoch: same as the unix epoch, i.e., 1970-01-01T00:00:00Z.
From that point, start counting seconds in successive prime-numbered sequence; these are termed 'dinosecs', such that epoch + 2s = dinosec 1, epoch + 2s + 3s = dinosec 2, epoch + 2s + 3s + 5s = dinosec 3, dinosec 3 + 7s = dinosec 4, etc etc.
It is, however, forbidden to use seconds in the actual specification of time - all time must be referred to in dinosecs and fractional dinosecs.
Here's my question: What is a good way to go about defining fractional dinosecs?
One obvious way is to simply take the number of seconds since the last dinosec and divide it by the number of seconds between that dinosec and the next one. I dislike that method because it makes dinosecs nondifferentiable at whole-numbers - i.e. there's a sudden change in how quickly the fractional part is incrementing.
Is there something that could be done with $x/\ln(x)$, for example? Or is there an more rigorous way to do it?
Any ideas would be appreciated.