Talk:Freestyle perfect numbers

I think that the given definition of freestyle perfect numbers is probably the most relaxed one which can be imagined. However, there are several variants possible to restrict what can be considered as spoof perfect number; all those cited below are compatible with Descarte's number:

1. Only one "spoof prime" is allowed.
2. The spoof prime factor(s) must be larger than all other factors. (Variants: larger than the prime factors, larger than any product of the true prime factors (or: smaller true or spoof prime factors) [with / without multiplicity].)
3. None of the smaller (spoof?) prime factors must divide the spoof prime factor(s).
4. Any spoof prime factor occurs always to the highest possible power, in the order of their size.(*)
5. Spoof primes cannot be even.
6. More generally, only P-spoof primes are allowed, which means they must not have a factor smaller than P. This P could be a function that grows with the size of the (potentially spoof perfect) number N. (Obviously P(N)=sqrt(N) is the limit beyond which one gets true primes.)

(*) An illustration: Consider the number N=(3*5)*(3*7)*(5*7). Without requirement 4, the spoof factorisation N=15*21*35 would be possible. But with requirement 4, the (so far) smallest factor 15 would be tested to higher powers, and since 15^2 divides, one would have the factorization N=7^2*15^2 (or rather N=15^2*49, taking 49 also as spoof prime, since by assumption the smallest factor found was 15, not 7). - — M. F. Hasler 03:55, 30 January 2013 (UTC)

Program

Any takers, to propose some code for producing such numbers? — M. F. Hasler 17:43, 30 January 2013 (UTC)