Problem Statement

Write a function which says whether a number is perfect.

A perfect number is a positive integer that is the sum of its proper positive divisors excluding the number itself. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors (including itself).

Note:   The faster   Lucas-Lehmer test   is used to find primes of the form   2n-1,   all known perfect numbers can be derived from these primes using the formula   (2n - 1) × 2n - 1. It is not known if there are any odd perfect numbers (any that exist are larger than 102000). The number of   known   perfect numbers is   51   (as of December, 2018),   and the largest known perfect number contains  49,724,095  decimal digits.

Let's start with the solution:

MODULE PerfectNumbers;

FROM SWholeIO IMPORT
  WriteCard;
FROM STextIO IMPORT
  WriteLn;
FROM RealMath IMPORT
  sqrt;

VAR
  N: CARDINAL;

PROCEDURE IsPerfect(N: CARDINAL): BOOLEAN;
VAR
  S, I: CARDINAL;
  SqrtN: REAL;
BEGIN
  S := 1;
  SqrtN := sqrt(FLOAT(N));
  IF N REM 2 = 0 THEN
    S := S + 2 + N / 2;
  END;
  I := 3;
  WHILE FLOAT(I) <= SqrtN - 1.0 DO
    IF N REM I = 0 THEN
      S := S + I + N / I;
    END;
    I := I + 1;
  END;
  IF I * I = N THEN
    S := S + I;
  END;
  RETURN (N = S);
END IsPerfect;

BEGIN
  FOR N := 2 TO 10000 BY 2 DO
    IF IsPerfect(N) THEN
      WriteCard(N, 5);
      WriteLn;
    END;
  END;
END PerfectNumbers.