A100832: What exactly are amenable numbers?

[David Wilson]

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