greedy computation of pseudoperfect numbers?

[Joerg Arndt]

For both sorts of pseudoperfect numbers, allowing divisor one A005835,
and forbidding divisor one A136446, a greedy method _seems_ to always
work for deciding whther a given n in in the sequence:

greedy(n, d)= \\ d:= vector of allowed divisors, sorted
{
    local(t=n,l=1);
    forstep (j=#d, 1, -1,
        if (t-d[j]>=0,
            t-=d[j];
            l = lcm(l, d[j]);
            print1(" +", d[j]);
            if (0==t,
                print(" yes,  lcm==n: ", l==n);
                if ( l!=n,  error(" lcm=!=n ") );
                return(1);
            );
        )
    );
    print(" no ");
    return( 0 );
}

The computation is obviously extremely fast.

Further, the lcm of all divisors taken for the sum has always lcm==1.
(cf. the line if ( l!=n,  error(" lcm=!=n ") ); which is never executed).

Is there any information available about whether the greedy approach
always works, and whether the lcm property holds?
I do not know whether the questions are easy or very hard.


Moreover, (disallowing the divisors one,) can any odd number n be in
sequence A136446?  This is of interest because all gcd computations in
a Rabin-like irreducibility test for polynomials over finite fields
(have to write it up) can be avoided for all numbers _not_ in A136446.

The "gcd-avoiding" numbers n are actually such that there is a multiset
of divisors that adds to n and has lcm==n;  that's why I ask about the
lcm property.