factors

Let P (n) denote the number of factors of n. Find all values of  n which satisfy P(n) = n.    

n =   1     1 factor:    1    

n =   2     2 factors:  1, 2    

n =   3     2 factors    

n =   4     3 factors:  1, 2, 4    

n =   5     2 factors    

n =   6     4 factors:  1, 2, 3, 6    

n =   7     2 factors    

n =   8     4 factors:  1, 2, 4, 8    

n =   9     2 factors    

n = 10     4 factors:  1, 2, 5, 10    

n = 11     2 factors   

n = 12     6 factors:  1, 2, 3, 4, 6, 12    

Those two are all I can think of.  As n gets larger, P(n) sometimes gets larger,     

but I think the trending demonstrates that P(n) is never going to catch up to n.    

_.