Just some observations.

• Any integer natural number may be decomposed into a product of powers of primes: $n == do { my $prod = 1; $prod *= $p[$_]**$k[$_] for 0..$#p; $prod } where @p are prime and @k are positive integers.
• All proper divisors of $n are products of the form do { my $prod = 1; $prod *= $p[$_]**int rand($k[$_]+1) for 0..$#p; $prod } , and all such products are divisors.
• There are therefore factorial(sum( @k)) proper divisors of $n. ^*
• If $n is larger than the sum of all its divisors, no subset can sum to it either.
• If $n is odd, an odd number of divisors will be in a sum which equals $n. If $n is even, an even number of the odd divisors will be in the sum.

You are right that this is a hard problem. The factorization of $n alone is an age-of-the-universe problem for large $n. Math::Pari is able to factor numbers of up to 60 digits fairly quickly. It is fast for less than 50 digits.

^* Oops, that is not quite right. If an element of @k is greater than 1, there is a binomial coefficient of ways to forms a power of that prime. All those are equivalent, yielding the same factor. That makes the factorial function I leaped at significantly larger than the actual number of distinct factors. The correct number of distinct divisors of $n is do { $prod = 1; $prod *= ($_ + 1) for @k; $prod } . Thanks to OverlordQ for pointing me to the right calculation.

After Compline,
Zaxo

Math::Pari (or its command-line interface gp) also has a function sigma(x) that computes the sum of all divisors of x, including itself. So it would be easy to use it to compute whether x is abundant, deficient, or perfect:

gp
? x = 168415508261
%1 = 168415508261
? sigma(x) - 2*x
%2 = -75992596042
[download]

If the result is less than zero, the number is deficient. If greater than zero, it's abundant. If zero, it's perfect.

That still doesn't find out for you if the number is weird or semiperfect, but it helps a little. Pari/gp can work with unlimited-precision numbers.

Update: Since the abundance (sigma(x) - 2x) can be computed, one possible search you could do is for odd numbers with a low abundance. Then you only need to deal with a small subset of the divisors, those smaller than the abundance. That set might be small enough to make a solution feasible.

For example, the abundance of 70 is 144-140=4. The only divisors that are under 4 are 1 and 2. Since there is no combination adding to 4, 70 is weird.

Update2: Pari/gp also has operations (e.g algdep, lindep, matkerint) for LLL lattice reduction, a powerful method that can often break down knapsack problems in polynomial time. Here is a reference to a paper on using lattice reduction to solve knapsack problems (weird number testing is basically a knapsack problem): Lattice Reduction: a Toolbox for the Cryptanalyst.

The Page used for that. A number like 1000 has prime factors 2^3 x 5^3.

Take the factor 2 = 0 1 2 3 times = 4 choices
Take the factor 4 = 0 1 2 3 times = 4 choices

So all together there could be 4*4 = 16 distinct factors.
Thanks to Zaxo for making me google for that :)

There is some analysis of this problem in this page:

http://www.cis.udel.edu/~breech/contest/problems/thoughts - see "Problem 2"

The interesting part is: "You only need to go to sqrt(x). If you find a d such that x%d == 0, then you know that d and x/d are both divisors. You do have to make a quick check to make sure that d and x/d are not equal before summing both".

And for the heck of it if you need test cases, here's the first few odd Semiperfect/Pseudoperfect numbers:

6, 12, 18, 20, 24, 28, 30, 36, 40, 42, 48, 54, 56, 60, 66, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 108, 112, 114, 120, 126, 132, 138, 140, 144, 150, 156, 160, 162, 168, 174, 176, 180, 186, 192, 196, 198, 200, 204, 208, 210, 216, 220, 222, 224, 228, 234, 240, 246, 252, 258, 260, 264

Ah! I wasn't sure how sophisticated you were or how much so you wanted to get. Now, I see you know what you want.

Since you don't care what $n is, you may be able to select a set of factors to obtain particular properties. I think that you will have better success going at the problem from an algebraic point of view than searching numerically. These kinds of problems have usually been mined out for many digits.

After Compline,
Zaxo

I think you meant to say "no odd wierd number is known." The reference Weird Number lists several even ones.

This reference to Semiperfect Numbers says that every multiple of a semiperfect number is semiperfect, which makes me think some form of sieving process would eliminate many cases quickly.

Update: I found a paper using google: Sums of Divisors and Egyptian Fractions which has some interesting results on weird numbers. Two things it mentioned are that a computer search proved there are no odd wierds below 2^32, and that all abundant numbers of the form 3^a * 5^b * 7^c are semiperfect (where a, b, c > 0).

CS people are notoriously mediocre mathemeticians. Why don't you find a math list to ask math questions

--
TTTATCGGTCGTTATATAGATGTTTGCA

On the other hand, sometimes mathematicians are mediocre programmers. Look at this "solution" to the subset sum problem: Subset Sum. He says to multiply out N binomials and collect the terms, which is O(2^N). Mathematically it's correct; computationally it's not feasible.

(I know that the subset sum problem is NP-complete, it's just the author of the article presents this as a solution without mentioning the issue at all.)