Project Euler Problem 45 Common Lisp

Project Euler Problem 45

Note: all hexagonal numbers are triangle numbers.  The function typep is used to see if the formula in the function pentagonal? is a natural number (integer).  Function problem45 increments up from 144, because the problem statement for Project Euler 45 already provides a hexagonal number 40755 (that's also a pentagonal and triangle number) using the integer 143.

(defun pentagonal? (num)
  (typep
    (/ (1+ (sqrt (1+ (* 24 num)))) 6)
    'integer))

(defun problem45 ()
  (loop for i upfrom 144
        for hex = (- (* 2 i i) i)
        when (pentagonal? hex) return hex))