trough@ihuxi.UUCP (Chris Scussel) (12/05/85)
A number is perfect if sigma(n) = 2*n. An alternate definition is when
s(n) = n, where s(n) is the sum of the divisors of the number, including 1
but not including n. Let's consider this definition "unfair"; 1 should be
excluded too, because
1) if k is a factor of n then n/k should be too
2) if k is a factor of n then k itself is the product of primes
unless k is 1
Even if the above reasons aren't very convincing, they lead to an interesting
question:
Are there any numbers such that s(n)-1 = n? (That is,
sigma(n)-1 = 2*n).
This is similar to another posted problem: s(x)-x = n, where n=-1.
Chris Scussel
AT&T Bell Labs
ihnp4!ihuxi!troughgclark@utcsri.UUCP (Graeme Clark) (12/08/85)
In article <1283@ihuxi.UUCP> trough@ihuxi.UUCP (Chris Scussel) writes: >A number is perfect if sigma(n) = 2*n. An alternate definition is when >s(n) = n, where s(n) is the sum of the divisors of the number, including 1 >but not including n. > > Are there any numbers such that s(n)-1 = n? (That is, > sigma(n)-1 = 2*n). We were interested in just this question. We called a number n such that s(n)=n+k a k-perfect number. Hence you are asking about 1-perfect numbers. We wrote a simple program to examine the numbers from 1 to 100,000 looking for k-perfect numbers for k in the range -100 to 100, and found some curious results: There were no 1-perfect numbers. All (-1)-perfect numbers were powers of two (it's clear that all powers of two are (-1)-perfect, but I don't know about the converse). For most values of k there were not very many (on the order of 10) k-perfect numbers in the range 1-100,000, but for two particular values of k there were (12 and 53 I think) there were lots and lots of k-perfect numbers (on the order of 1000). Does anybody know anything that would shed some light on these results? Graeme Clark ihnp4!utcsri!gclark
franka@mmintl.UUCP (Frank Adams) (12/10/85)
In article <1741@utcsri.UUCP> gclark@utcsri.UUCP (Graeme Clark) writes: >We were interested in just this question. We called a number n such >that s(n)=n+k a k-perfect number. Hence you are asking about 1-perfect >numbers. We wrote a simple program to examine the numbers from 1 to 100,000 >looking for k-perfect numbers for k in the range -100 to 100, and found >some curious results: > > There were no 1-perfect numbers. > > All (-1)-perfect numbers were powers of two (it's clear that all > powers of two are (-1)-perfect, but I don't know about the converse). > > For most values of k there were not very many (on the order of 10) > k-perfect numbers in the range 1-100,000, but for two particular > values of k there were (12 and 53 I think) there were lots and lots > of k-perfect numbers (on the order of 1000). I think it must have been 12 and 56. Generally, if N is a perfect number, and p is prime which does not divide N, then pN is a (2N)-perfect number. (This is a matter of simple arithmetic.) Generally, one would expect k-perfect numbers with k odd to be rather rare. Note that n is k-perfect if sigma(n)-2n = k. (sigma is the sum of the divisors function.) This can be odd only if sigma(n) is odd. The only prime powers with sigma(n) odd are (1) powers of two, and (2) even powers of odd primes. Since sigma is multiplicative (if a and b have no common factors, sigma(ab)=sigma(a)sigma(b)), only numbers of the form (2^n)(u^2), where u is odd, will have odd values of sigma. For the cases k = 1 and k = -1, the constraints are rather severe. We are trying to solve sigma(n) = 2n + 1 or 2n - 1. This means that n and sigma(n) must have no common factors. Since numbers with sigma(n) close to 2n must have a fair number of factors, and therefore sigma(n) will also have a fair number of factors, it is hard for this to be the case. It is therefore a likely conjecture that these equations have no solutions except for powers of two. I don't really see how to start constructing a proof, however. I would guess that analytic methods would be required. Frank Adams ihpn4!philabs!pwa-b!mmintl!franka Multimate International 52 Oakland Ave North E. Hartford, CT 06108