Next number is 169.

First number 83 -> 8+3=11 and square of 11 is 121

Second number 121 -> 1+2+1 = 4 and square of 4 is 16

Thirst number 16 -> 1+6 = 7 and square of 7 is 49

Fourth number 49 -> 4+9 = 13 and square of 13 is 169

Fifth number 169 -> 1+6+9 = 16 and square of 16 is 256

Sixth number 256 -> 2+5+6 = 13 and square of 13 is 169