In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself.
The sum of divisors of a number, excluding the number itself, is called its It is not known whether there are any odd perfect numbers, nor whether infinitely many perfect numbers exist. Seems like a very difficult problem for a first-time user.You might want to recheck the statement of the problem. 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. The discovery of such numbers is lost in prehistory, but it is known that the Pythagoreans (founded c. 525 BCE) studied perfect numbers for … This test program applies the function to a list of the first five hundred natural (where the divisors adds upto to the number. *//*──────────────────────────────────────────────────────────────────────────────────────*//*REXX program tests if a number (or a range of numbers) is/are perfect.
the quoted example is 28[1,2,4,7,14])
This program tests 10**2001 in about 30 seconds - but only for even perfects. 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 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 A number is said to be anti-perfect if it is equal to the sum of the reverses of its proper divisors.
In addition to generating even perfect numbers, we can also have a fast function which returns true when a given even number is perfect: A short intro to PROC FCMP is at Finding the first 6 of just about anything is likely to be much easier than an arbitrary N as indicated in your first post.
The first few perfect numbers are For example, the first four perfect numbers are generated by the formula 2Four higher perfect numbers have also been discovered, namely those for which It is unknown whether there is any odd perfect number, though various results have been obtained.
The first 38 in about 1 second with most of the time printing the large results. With 31 bit integers the limit is 2,147,483,647. Every piece is 100% handmade in Los Angeles. Not all p of this form lead to perfect totient numbers; for instance, 51 is not a perfect totient number.
I don't know if your class discussed PROC FCMP, which enables you to define DATA step functions. Thank you
Alternatively, a more efficient check for even perfect numbers: Yes pretty difficult. Write a function which says whether a number is perfect. Examples (testing 496, testing 128, finding all perfect numbers in 1...10000): Also, it has been conjectured that there are no odd Ore's harmonic numbers. 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. Euclid saw that 2 n - 1 is a prime number in these four cases. The known anti-perfect numbers are 6, 244, 285, 133857, 141635817
The next in line are 496, 8128, and 33550336. This version uses the Lucas-Lehmer method, digital roots, and restricts itself to An integer square root function was added to limit the factorization of a number.
Examples of use, including extensions beyond those assumptions: For example, 244 is anti-perfect since its proper divisors are 1, 2, 4, 61, 122 and the sum of their reversal is 1 + 2 + 4 + 16 + 221 = 244. Both ntheory and Math::Pari have useful functions for this. A common programming assignment is to determine whether a number is a perfect This is much easier than determining whether a number is perfect, which is an abstract property in number theory.Okay.
Perfect numbers that are even. More generally if p is a prime number greater than 3, and 3p is a perfect totient number, then p ≡ 1 (mod 4) (Mohan and Suryanarayana 1982). Euclid saw that 2 n - 1 is a prime number in these four cases. (Monadic return/inject for lists is simply lambda x --> [x], inlined here, and fail is [].) (sequence A005537 in OEIS). 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. Faster tests would use the LL test for evens and myriad results on OPNs otherwise.
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