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Jan 30, 2000, 3:00:00 AM1/30/00

to

As I understand it, a friendly number is a number, under a function that

sums the factors of the number and then divides by the number is equal to

another number which has the same solution.

sums the factors of the number and then divides by the number is equal to

another number which has the same solution.

For example:

6 and 28 are friendly numbers because they both total 2 when the function is

applied.

I would like to know what is the purpose of such a term and function?

-Larry

Jan 31, 2000, 3:00:00 AM1/31/00

to

Lawrence Allie <la...@allie.com> a écrit dans le message :

38947...@news3.prserv.net...

> As I understand it,

Misunderstanding ,alas

a friendly number is a number, under a function that

> sums the factors of the number and then divides by the number is equal to

> another number which has the same solution.

Strange way of putting it, but the example below is clearer

>

> For example:

>

> 6 and 28 are friendly numbers because they both total 2 when the function

is

> applied.

No: they are "perfect", and not related otherwise."Friendly" is the pair

(220,284)

>

> I would like to know what is the purpose of such a term and function?

>

Hard to answer, that one... Would we say "to satisfy human curiosity (about

integers and their divisors)"?

> -Larry

>

>

Jan 31, 2000, 3:00:00 AM1/31/00

to

Lawrence Allie <la...@allie.com> wrote:

> As I understand it, a friendly number is a number, under a function that

> sums the factors of the number and then divides by the number is equal to

> another number which has the same solution.

>

> For example:

>

> 6 and 28 are friendly numbers because they both total 2 when the function is

> applied.

>

> I would like to know what is the purpose of such a term and function?

Denis Feldmann <denis.f...@wanadoo.fr> replied:

> No: they are "perfect", and not related otherwise."Friendly" is the pair

> (220,284)

Actually, "amicable" is the preferred term for a pair like (220,284),

in which sigma(m)-m=n and sigma(n)-n=m. "Friendly pairs" were defined,

roughly as Lawrence stated, in:

Claude Anderson and Dean Hickerson, Problem 6020,

Amer. Math. Monthly 84 (1977) 65-66

Specifically, if sigma(m)/m = sigma(n)/n and m != n, then m and n are

said to be a friendly pair. (Here sigma(n) is the sum of the positive

divisors of n.)

Generally, we were interested in determining, for each rational number

r>1, how many positive integers n satisfy the equation sigma(n)/n = r.

Assuming that there are infinitely many Mersenne primes, then there are

infinitely many even perfect numbers, so for r=2 there are infinitely

many solutions. More generally, if k is an odd number and r = 2 sigma(k)/k,

then there are infinitely many even perfect numbers P which are relatively

prime to k. For such a P, let n = kP; then sigma(n)/n = r. So for

such values of r there are infinitely many solutions.

I conjecture that for all other values of r there are only finitely many

solutions. For certain values of r it's easy to prove that there are

no solutions; e.g. r=5/4. For others it's easy to prove that there's

exactly one; e.g. r=3/2. But for some values of r it seems hopeless to

determine all of the solutions. E.g. for r=7/3, we may take n=12 or 234,

and I doubt that there are any other solutions, but I also doubt that

anyone will ever prove that.

I think it was Carl Pomerance who found essentially the only known

example of an r for which we can prove that there are finitely many

solutions, but more than one. Specifically, he found that for r=93/40,

the only solutions are n=80 and n=200. Other such values of r can be

constructed from this one, like r=93(p+1)/(40p) where p is a prime

such that p+1 is not divisible by 4, 5, or 7. I'd very much like to

know if there are any values of r for which we can prove that there are

finitely many values of n, but more than one, other than those based on

the 93/40 example.

Dean Hickerson

de...@math.ucdavis.edu

Feb 1, 2000, 3:00:00 AM2/1/00

to

Lawrence Allie wrote:

>As I understand it, a friendly number is a number, under a function

>that sums the factors of the number and then divides by the number

>is equal to another number which has the same solution.

>As I understand it, a friendly number is a number, under a function

>that sums the factors of the number and then divides by the number

>is equal to another number which has the same solution.

>For example:

>6 and 28 are friendly numbers because they both total 2 when the

>function is applied.

>I would like to know what is the purpose of such a term and function?

To give the specific definition that the previous poster implied:

Two numbers are "friendly" if the sum of their factors (NOT including

the number itself are equal.

A number is "perfect" if the sum of its factors (again, not

including itself) is equal to itself. Both 6 and 28 are perfect: 6

has factors 1, 2, 3, 6. 1+2+3= 6. 28 has factors 1, 2, 4, 7, 14, 20.

1+ 2+ 4+ 7+ 14= 28.

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