*//*──────────────────────────────────────────────────────────────────────────────────────*//*REXX program tests if a number (or a range of numbers) is/are perfect. Euclid saw that 2 n - 1 is a prime number in these four cases. An example is the number 6. So which is it, first 6 or first N?Given the limitations of SAS, only the first 7 perfect numbers can be calculated with numeric variables anyway (see If true, this would mean that there are no odd perfect numbers. 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. A005100), where sigma(n) is the sum of the divisors of n (). Not all p of this form lead to perfect totient numbers; for instance, 51 is not a perfect totient number. Programming note: this traditional method takes advantage of a few shortcuts: ...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. Takes about 15 seconds: Perfect numbers have the property that the integer divisors sum to the number.
A005101), perfect if sigma(n) = 2n (this entry), deficient if sigma(n) < 2n (cf. Caveat: this goes well past the current bound for odd perfect numbers and does not check for them. Auto-suggest helps you quickly narrow down your search results by suggesting possible matches as you type. Write code in SAS that allows one to write to a csv file a data set with all natural numbers less than a given parameter as well as a boolean variable indicating if the number is perfect or not. Taking gives the usual perfect numbers.. Iannucci et al. Yes pretty difficult. In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. Numbers where the sum is less than the number itself are called Commentary on the Gospel of John 28.1.1-4, with further references in the The Collected Mathematical Papers of James Joseph Sylvester p. 590, tr. from "Sur les nombres dits de Hamilton", The first few numbers in the sequence of k-hyperperfect numbers are 6, 21, 28, 301, 325, 496, 697, ... (sequence A034897 in the OEIS), with the corresponding values of k being 1 If is an odd integer, and and are prime, then is -hyperperfect.McCranie (2000) conjectures that all -hyperperfect numbers for odd are in fact of this form. *//*──────────────────────────────────────────────────────────────────────────────────────*//*REXX program tests if a number (or a range of numbers) is/are perfect.
The sum of the divisors is 6. It was known to Euclid that if is prime, then is a perfect number. Alternatively, a more efficient check for even perfect numbers: Also, it has been conjectured that there are no odd Ore's harmonic numbers. It requires us to get first N (6) perfect numbers. *//*──────────────────────────────────────────────────────────────────────────────────────*/13164036458569648337239753460458722910223472318386943117783728128 As usual, this is easier and faster with modules. Euclid saw that 2 n - 1 is a prime number in these four cases. and two ASSIST macros (XDECO,XPRNT) to keep it as short as possible. Every piece is 100% handmade in Los Angeles. Uses built-in method. Perfect numbers that are even. This REXX version makes use of the fact that all This version uses memoization to implement a fast version of the Lucas-Lehmer test. (where the divisors adds upto to the number. In a 1638 letter to Between Euclid and Euler, medieval mathematicians made some conjectures about perfect numbers that have since been proven false, such as that there is a perfect number between each consecutive power of 10 (there are no perfect numbers between 10000 and 100000, or for that matter between 10000 and 10000000), and that the least significant base 10 digit of each successive perfect number alternates between 6 and 8 (supported by reference only to the first five perfect numbers).Follow the thread relating to the following SeqFan post on The divisors are 1, 2 and 3. But the question is correct. Various results have been obtained, but none that has helped to locate one or otherwise resolve the question of their existence. See also Rational Arithmetic Perfect numbers on OEIS Odd Perfect showing the current status of bounds on odd perfect numbers. Both ntheory and Math::Pari have useful functions for this. *//*──────────────────────────────────────────────────────────────────────────────────────*//*REXX program tests if a number (or a range of numbers) is/are perfect.