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.

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