A100832: What exactly are amenable numbers?

[David Wilson]

Actually, Eric Weisstein is known to frequent the math-fun and seqfan
mailing lists, and we, myself included, have contributed much to his
encyclopedia.  I would not be surprised if he picks up on this and
makes the suggested changes.

----- Original Message ----- 
From: "Alonso Del Arte" <alonso.delarte at gmail.com>
To: <ham>; "David Wilson" <davidwwilson at comcast.net>
Sent: Sunday, January 09, 2005 2:19 PM
Subject: Re: A100832: What exactly are amenable numbers?


> When you said "a multiset S of exactly n integers" that cleared up my
> confusion. It says so in the sequence, algebraically, but I didn't
> realize it at first.
>
> There probably isn't anything we can do about the Mathworld article,
> but the Wikipedia article can be edited by anyone, so I have changed
> it to call S a multiset.
>
> Alonso
>
>
> On Fri, 7 Jan 2005 21:32:34 -0500, David Wilson
> <davidwwilson at comcast.net> wrote:
>> I looked at both the Mathworld and Wikipedia articles.  From the examples
>> in both, I understand an amenable number to be a positive integer n which 
>> is
>> both the sum and product of a single multiset S of two or more positive
>> integers.  S must be a multiset, otherwise our sum or product cannot 
>> involve
>> repeated terms, as it clearly does.  S must have two or more elements,
>> otherwise all positive numbers n are amenable with S = {n}.  By this
>> definition, the amenable numbers are precisely the positive composite
>> numbers.
>>
>> Oddly, neither Mathworld nor Wikipedia has a precise explicit definition.
>> Mathworld is unclear that S is a multiset and omits that S must have two 
>> or
>> more elements.  Wikipedia calls S a set, when it is in fact a multiset.
>>
>> A100832 is a variant on amenable numbers where we allow S to contain
>> negative integers and require that |S| = n, in other words, n is both the
>> product and sum of a multiset S of exactly n integers.
>>
>> Empirically, I have verified that, up to 1000, A100832 include precisely
>> the numbers == 0 or 1 (mod 4), except for 4.  For each n of this form,
>> we can compute the associated S as follows:
>>
>> S = { n/2, 2, 1 x (3n/4-2), -1 x (n/4) }  if n == 0 (mod 8).
>> S = { n/2, -2, 1 x (3n/4), -1 x (n/4-2) } if n == 4 (mod 8) except n = 4.
>> S = { n, 1 x (n-1)/2, -1 x (n-1)/2 } if n == 1 (mod 4).
>>
>> I suspect there is probably a fairly straightforward proof that no such S
>> exists for n == 2 or 3 (mod 4), but it's a little beyond me at the 
>> moment.
>>
>> ----- Original Message -----
>> From: "Alonso Del Arte" <alonso.delarte at gmail.com>
>> To: <ham>; <seqfan at ext.jussieu.fr>
>> Sent: Friday, January 07, 2005 5:40 PM
>> Subject: A100832: What exactly are amenable numbers?
>>
>> > The sequence of amenable numbers, A100832, was added today. I looked
>> > to Mathworld for more information, there it says that all composites
>> > are amenable. Wikipedia elaborates that
>> >
>> > "One can always make an inelegant solution by taking the prime
>> > factorization (expressed with repeated factors rather than exponents)
>> > and add as many 1s as necessary to add up to n. Because of the
>> > multiplicative identity, multiplying this set of integers will yield n
>> > no matter how many 1s there are in the set."
>> >
>> > Yet, some composites are not in the sequence, 4, 6, 10, 14, (in short,
>> > all composites not congruent to 0 or 1 mod 4).
>> >
>> > Wikipedia even goes on to say that any prime number can be amenable
>> > with the set {1, -1, 1, -1, p}.
>> >
>> > So what are the constraints on amenable numbers? Is the congruence
>> > required? Are negative numbers allowed in the sets?
>> >
>> > Alonso
>>
>>