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For property s, I have found multiples of 1, 2 and 9 - not 4 as stated in preamble.

Thanks Rad, after some mind wrenching it suddenly dawned on me that the number giving the grid entry number is not necessarily the same number that leads to the grid entry. Once I realised that, there are only 3 possible answers for property e, those that you cite. I have now actually started to fill the grid (only 3 answers), but progress is progress!! I now need some fresh air and will tackle this monster later with a clear mind.

Ignore that - I was wrong about 2: 4 is OK.

Dryden

9 is 3 times 3, so that is two factors 3, an even number of 3s.

Rad,

There does appear to be a contradiction. 9 is permitted on the prime factorisation issue but excluded on the non-zero square issue.

Dryden

The contradiction is caused by the incorrect helpful information given. 9 and 36 are clearly not allowed for property f. I have just realised that 64 is not allowed either, so we have to be wary of perfect squares, though 25 would be OK.

Is this the explanation?
9 has two identical prime factors, 3 and 3. Each is one less than a multiple of 4 and there is an odd number of each, so 9 is not the sum of two squares.

Rad's first post appears to be correct, 'non-zero' should be omitted in the preamble.

Dryden

By that reasoning every factor of every number would be unique, in a class of its own, and there would never be two of the same!

Sheer hard slog, but at least all that's required is basic arithmetic. I have it complete apart from 37 where the first digit is unchecked. I reckon it has to be a prime number, but there are 3 primes which would fit. Has anyone else encountered this, or do I have to flog through all the other criteria to make sure that none of them is accidentally satisfied as well?