The number of known perfect numbers is 50 (as of September, 2018), and the largest known perfect number contains over 46 million decimal digits. See also Rational Arithmetic Perfect numbers on OEIS Odd Perfect showing the current status of bounds on odd perfect numbers. from "Sur les nombres dits de Hamilton", While many of Euclid's successors implicitly assumed that all perfect numbers were of the form (Dickson 2005, pp. A005100), where sigma(n) is the sum of the divisors of n (). Examples of use, including extensions beyond those assumptions: Seems like a very difficult problem for a first-time user.You might want to recheck the statement of the problem. All end in 6 or 8, though what seems to be an alternating pattern of 6's and 8's for the first few perfect numbers doesn't continue.

Iannucci et al. The first 20 takes about a second. (Monadic return/inject for lists is simply lambda x --> [x], inlined here, and fail is [].) *//*──────────────────────────────────────────────────────────────────────────────────────*//*REXX program tests if a number (or a range of numbers) is/are perfect.
As usual, this is easier and faster with modules. *//*──────────────────────────────────────────────────────────────────────────────────────*//*REXX program tests if a number (or a range of numbers) is/are perfect. *//*──────────────────────────────────────────────────────────────────────────────────────*/13164036458569648337239753460458722910223472318386943117783728128 If so, I recommend that you write an FCMP function that takes a number and computes whether it is perfect by finding the sum of all integer divisors. The smallest perfect number is 6, which is the sum of 1, 2, and 3. ...a prolonged meditation on the subject has satisfied me that the existence of any one such [odd perfect number] — its escape, so to say, from the complex web of conditions which hem it in on all sides — would be little short of a miracle.All even perfect numbers have a very precise form; odd perfect numbers either do not exist or are rare. A perfect number is a natural number or a positive integer whose proper divisors sum up to the same number. With 31 bit integers the limit is 2,147,483,647. For example: [math]6[/math] is the first perfect number, since [math]6[/math] has the factors [math]1,2[/math] and [math]3[/math]. N > 10300. Or we can be clever and look for 2^(p-1) * (2^p-1) where 2^p -1 is prime. This REXX version makes use of the fact that all   This version uses memoization to implement a fast version of the Lucas-Lehmer test. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors (including itself). Examples (testing 496, testing 128, finding all perfect numbers in 1...10000): the quoted example is 28[1,2,4,7,14]) Finding the first 6 of just about anything is likely to be much easier than an arbitrary N as indicated in your first post. In it he writes about perfect numbers and amicable numbers. A005101), perfect if sigma(n) = 2n (this entry), deficient if sigma(n) < 2n (cf. Any odd perfect number Nmust satisfy the following conditions: 1. The output is the same as for the ooRexx version (above). The ancient Christian scholar Augustine explained that God could have created the world in an instant but chose to do it in a perfect number of days, 6. An even number is perfect if and only if it equals 2 k - 1 ⁢ ( 2 k - 1 ) for some integer k > 1 and 2 k - 1 is prime. *//*──────────────────────────────────────────────────────────────────────────────────────*//*REXX program tests if a number (or a range of numbers) is/are perfect.