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Friendly number

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Lawrence Allie

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.

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

denis.feldmann

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
>
>

Dean Hickerson

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?

> 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

G.E. Ivey

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.

>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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