Problem:
Score: 200 points

Problem Statement

The number 105 is quite special - it is odd but still it has eight divisors.
Now, your task is this: how many odd numbers with exactly eight positive divisors are there between 1 and N (inclusive)?

Constraints

N is an integer between 1 and 200 (inclusive).

Input

Input is given from Standard Input in the following format:

N

Output

Print the count.

Sample Input 1

105

Sample Output 1

1

Among the numbers between 1 and 105, the only number that is odd and has exactly eight divisors is 105.

Sample Input 2

7

Sample Output 2

0

1 has one divisor. 3, 5 and 7 are all prime and have two divisors. Thus, there is no number that satisfies the condition.

Solution in Common Lisp (SBCL 2.0.3) (judge verdict: Accepted)

(in-package :cl-user)
(eval-when (:compile-toplevel :load-toplevel :execute)
  (sb-int:defconstant-eqx opt
    #+swank '(optimize (speed 3) (safety 2))
    #-swank '(optimize (speed 3) (safety 0) (debug 0))
    #'equal)
  #+swank (ql:quickload '(:cl-debug-print :fiveam) :silent t)
  #-swank (set-dispatch-macro-character
           #\# #\> (lambda (s c p) (declare (ignore c p)) `(values ,(read s nil nil t)))))
#+swank (set-dispatch-macro-character #\# #\> #'cl-debug-print:debug-print-reader)

(macrolet ((def-int (b)
             `(progn (deftype ,(intern (format nil "UINT~A" b)) () '(unsigned-byte ,b))
                     (deftype ,(intern (format nil "INT~A" b)) () '(signed-byte ,b))))
           (defs (&rest bits) `(progn ,@(mapcar (lambda (b) `(def-int ,b)) bits))))
  (defs 2 4 7 8 15 16 31 32 62 63 64))

(defconstant +mod+ 1000000007)

(defmacro dbg (&rest forms)
  #+swank (if (= (length forms) 1)
              `(format *error-output* "~A => ~A~%" ',(car forms) ,(car forms))
              `(format *error-output* "~A => ~A~%" ',forms `(,,@forms)))
  #-swank (declare (ignore forms)))

(declaim (inline println))
(defun println (obj &optional (stream *standard-output*))
  (let ((*read-default-float-format* 'double-float))
    (prog1 (princ obj stream) (terpri stream))))

;; BEGIN_INSERTED_CONTENTS
(defpackage :cp/divisor
  (:use :cl)
  (:export #:enum-divisors #:enum-ascending-divisors #:make-divisors-table))
(in-package :cp/divisor)

(declaim (ftype (function * (values (vector (integer 0 #.most-positive-fixnum)) &optional))
                enum-divisors))
(defun enum-divisors (x)
  "Enumerates all the divisors of X in O(sqrt(X)) time. Note that the resultant
vector is NOT sorted."
  (declare (optimize (speed 3))
           ((integer 0 #.most-positive-fixnum) x))
  (let* ((sqrt (isqrt x))
         (result (make-array (isqrt sqrt) ; FIXME: currently set the initial size to x^1/4
                             :element-type '(integer 0 #.most-positive-fixnum)
                             :fill-pointer 0)))
    (loop for i from 1 to sqrt
          do (multiple-value-bind (quot rem) (floor x i)
               (when (zerop rem)
                 (vector-push-extend i result)
                 (unless (= i quot)
                   (vector-push-extend quot result)))))
    result))

;; Below is a variant that returns a sorted list.
(defun enum-ascending-divisors (n)
  "Returns the ascending list of all the divisors of N."
  (declare (optimize (speed 3))
           ((integer 1 #.most-positive-fixnum) n))
  (if (= n 1)
      (list 1)
      (let* ((sqrt (isqrt n))
             (result (list 1)))
        (labels ((%enum (i first-half second-half)
                   (declare ((integer 1 #.most-positive-fixnum) i))
                   (cond ((or (< i sqrt)
                              (and (= i sqrt) (/= (* sqrt sqrt) n)))
                          (multiple-value-bind (quot rem) (floor n i)
                            (if (zerop rem)
                                (progn
                                  (setf (cdr first-half) (list i))
                                  (setf second-half (cons quot second-half))
                                  (%enum (1+ i) (cdr first-half) second-half))
                                (%enum (1+ i) first-half second-half))))
                         ((= i sqrt) ; N is a square number here
                          (setf (cdr first-half) (cons i second-half)))
                         (t ; (> i sqrt)
                          (setf (cdr first-half) second-half)))))
          (%enum 2 result (list n))
          result))))

(declaim (ftype (function * (values (simple-array list (*)) &optional))
                make-divisors-table))
(defun make-divisors-table (sup)
  "Returns a vector of length SUP whose each cell, vector[X], is the ascending
list of every divisor of X. Note that vector[0] = NIL."
  (declare ((integer 0 #.most-positive-fixnum) sup)
           #+sbcl (sb-ext:muffle-conditions style-warning))
  (let ((result (make-array sup :element-type 'list))
        (tails (make-array sup :element-type 'list))) ; stores the last cons cell
    (declare (optimize (speed 3) (safety 0)))
    (loop for i from 1 below sup
          for cell = (list 1)
          do (setf (aref result i) cell
                   (aref tails i) cell))
    (when (>= sup 1)
      (setf (aref result 0) nil))
    (loop for divisor from 2 below sup
          do (loop for number from divisor below sup by divisor
                   do (setf (cdr (aref tails number)) (list divisor)
                            (aref tails number) (cdr (aref tails number)))))
    result))

;; BEGIN_USE_PACKAGE
(eval-when (:compile-toplevel :load-toplevel :execute)
  (use-package :cp/divisor :cl-user))
(in-package :cl-user)

;;;
;;; Body
;;;

(defun main ()
  (let* ((n (read))
         (table (make-divisors-table (+ n 1))))
    (println (loop for i from 1 to n by 2
                   count (= (length (aref table i)) 8)))))

#-swank (main)

;;;
;;; Test and benchmark
;;;

#+swank
(defun get-clipbrd ()
  (with-output-to-string (out)
    #+os-windows (run-program "powershell.exe" '("-Command" "Get-Clipboard") :output out :search t)
    #+os-unix (run-program "xsel" '("-b" "-o") :output out :search t)))

#+swank (defparameter *this-pathname* (uiop:current-lisp-file-pathname))
#+swank (defparameter *dat-pathname* (uiop:merge-pathnames* "test.dat" *this-pathname*))

#+swank
(defun run (&optional thing (out *standard-output*))
  "THING := null | string | symbol | pathname

null: run #'MAIN using the text on clipboard as input.
string: run #'MAIN using the string as input.
symbol: alias of FIVEAM:RUN!.
pathname: run #'MAIN using the text file as input."
  (let* ((*standard-output* (or out (make-string-output-stream)))
         (res (etypecase thing
                (null
                 (with-input-from-string (*standard-input* (delete #\Return (get-clipbrd)))
                   (main)))
                (string
                 (with-input-from-string (*standard-input* (delete #\Return thing))
                   (main)))
                (symbol (5am:run! thing))
                (pathname
                 (with-open-file (*standard-input* thing)
                   (main))))))
    (if out res (get-output-stream-string *standard-output*))))

#+swank
(defun gen-dat ()
  (uiop:with-output-file (out *dat-pathname* :if-exists :supersede)
    (format out "")))

#+swank
(defun bench (&optional (out (make-broadcast-stream)))
  (time (run *dat-pathname* out)))

#-swank
(eval-when (:compile-toplevel)
  (when (and (boundp 'sb-c::*compiler-warning-count*)
             (> sb-c::*compiler-warning-count* 0))
    (sb-ext:quit :unix-status 1)))

;; To run: (5am:run! :sample)
#+swank
(it.bese.fiveam:test :sample
  (5am:is
   (equal "1
"
          (run "105
" nil)))
  (5am:is
   (equal "0
"
          (run "7
" nil))))