Consider a calender based on the number 7: 7 days in a week, 7 weeks in a month, 7 full months in a year. The benefit is it is fairly straightforward to calculate the day of the year: just convert from base 7.
An extra short month with 3 or 4 weeks is tacked on at the end of every year (so 364 or 371 days in a year). The leap week happens every 5 years. Omit the leap week every 40 years. Don't omit the leap week every 400 years. This happens to be exactly the same length as the 400-year Gregorian calendar cycle (146097 days).
All the extra weeks and leap weeks go at the end of the year, so it does not interfere too much with day-of-year calculations: just add or subtract 7^3=343.
A less radical way to accomplish the same benefit is 11 31-day months and the last month has 24 or 25 days. The year could begin in March, leaving February as the short month with a leap day. This avoids Christmas disappearing (if December were the short, last month). Conversion to day of year is simply base 31.