4153

5^4 + 11^3 + 13^3

4157 = (41^2) + (17^2) + (3^7)

4159

The largest known prime that is the mean of two consecutive Mersenne primes. (2^7-1+2^13-1)/2

2^9 + 3^4 + 13^3 + 37^2

4177 = (53^2) + (37^1) + (11^3)

4201

2^10 + 5^4 + 7^3 + 47^2

31

is not the next :s

But this is:
4211 = (59^2) + (3^6) + (2^0)

4217

11463/4217 (2.718283139) is a decent rational approximation to e.

2^9 + 7^3 + 7^4 + 31^2

4219 = (53^2) + (41^1) + (37^2)