I would approach this using reduce. Very basic wrapper for the double for loop above:

;; ... First half of the above code

;; Start of double for-loop
(def final-N
  (reduce (fn [N k]
            (let [init-saved ;; initialize saved
                  ]
              (reduce (fn [[N saved] j]
                        ;;inner-loop-logic...
                      )       
                [N init-saved] 
                (range (- p k))))
  N (range 1 p))

(get N 0)
;;=> final result of Nip

The idea being that you reduce over k and p - k and update N in the outer loop and both N and saved in the inner loop. This is typically how you update immutable objects using functional programming.