2013 and 2014 as integers to factor are fairly interesting. Each has three prime factors with no repeats. Each has one one digit prime factor, one 2 digit prime factor under 20 and one 2 digit prime factor over 30. It's a miracle!

2013 = 2 * 19 * 53

2014 = 3 * 11 * 61

and no factors in common. No repeated factors. 2015 follows completely in the footsteps of 2013 and 2014

2015 = 5 * 13 * 31

No repeated factors and no prime factors in common with 2013 or 2014!

It's not until we get to 2016 that the pattern is not just broken, but completely destroyed: 2 repeated factors and no factor over 7.

2016 = 2^⁵ * 3^² * 7

At least the powers 5 and 2 are also prime. And there are exactly 3 distinct prime factors, and 3 is prime.  

Today's quiz.

• What was the last prime year?
• What was the most recent year before 2013 that had the same properties of factorization as 2013, 2014 and 2015?  As this is ambiguous, try these versions of this question
  • Three prime factors, one under 10, one between 10 and 20, one over 30.
  • That, plus no prime factors in common with 2013, 2014 or 2015?
  • What is the next year that satisfies both these properties?
• What was the last year with exactly two factors?  (Note: three distinct correct answers, give both. )

Remember: there are 3 kinds of statisticians, those who can count, and those who can't.  

Todays relatively hard bonus quiz.  All questions refer to years AD prior to 2014.

• How many years with a prime number of prime factors where each prime factor's multiplicity is prime?
• How many with one prime factor?
  • Two distinct prime factors?
  • ​Three distinct prime factors?
• ​Repeat that last question, where each factor has multiplicity one.  

Meta bonus quiz question: What assumptions am I making that distinguishes the quiz from the relatively hard bonus quiz?

Happy new year!