The other day I started messing around with some properties and noticed a pattern emerging when the digit product and digit sum of a number were added together. For example, 15. (1+5)+(1*5) = 11. If you keep going you'll find a sequence that goes through every number from 1 to 18 exactly once.
Then if you go further you'll find that 19, and in fact any two-digit number ending in 9, refers to itself. (E.g. 59 = 5*9 + 5+9) You can prove this just by using two forms of these numbers: 10a + b = ab + a + b, which goes to 9a = ab, b=9.
Going further, I noticed that the form ab + a + b can be rewritten as (a+1)(b+1) - 1. So in general, above 19, the property of the number is one more than its tens digit times one more than its unit digit, minus one.
- 0 recalls itself
- 1 to 18 loop round in the sequence 1,2,4,8,16,13,7,14,9,18,17,15,11,3,6,125,10,1...
- 19 recalls itself
- 20 to 29 follow the form 3n - 1
- 30 to 39 follow the form 4n - 1 Etc...
I would like to know if there is any particular reason for the 1 to 18 sequence, or if there are any other loops, or if this sequence has a name.