The problem of designing a calendar with twelve month and 365 days is how to distribute the odd 5 days. The pattern seems to be consistent with using multiples of 30 7/12 rather than the more obvious 30 5/12 as seen in the calculation below. An irregularity is that the sequence is shifted by one month.
In order to do this one needs to be able to perform integer division and this can be done quite easily using multiplication tables and Sosigenes would have been quite capable of doing this. This gives us the sums of the days of the months. The formula for computing the sums turns out to be rather simple and allows us to derive a formula for January through December counting January as the first month. This simplifies converting month and day to day of year with the inclusion of a leap day in leap years.